Sometimes the past gives us actual puzzles to solve (or try to), like who killed the princes in the tower. More typically, historical research is a question of understanding something better: why a phenomenon occurred (f.e. the rise of the West), the short and long term effects of something, relationships between events.

From there, you can divide investigations/debates into a couple different groups. One set of research is largely methodological or theoretical. For example, the question of personal agency as a driver of historical events. Do the actions of individual people matter, or are we all mere pawns being swept around by the forces of Society? For example: how important was the specific person of Martin Luther to the initiation and success of a split in the Church in the sixteenth century. Or: who are the primary actors of History? Nation-states? The upper class rulers? Just men and the occasional woman in a position of legal authority, or also the women supporting them behind the scenes? The working classes? How historians tell long-term historical narratives is going to depend very much on their own answers to those questions.

But then there are the really meaty questions, major turning points in history whose causes, courses, and effects have not been fully charted. Here are a couple in my field, medieval/early modern, that are exciting to me:

(1) Basically everything surrounding the initial rise of Islam--how and why did it spread; who was a part of the movement; how and when did the Quran come to take the shape we know it today. This line of inquiry was exploded open by Patricia Crone, especially, a couple decades ago when scholars began to argue that we cannot accept the Quran and the hadiths as uncritical, factual accounts of early Islam. Today, Robert Hoyland and Fred Donner are two of the most important scholars publishing on this topic, with some different opinions. You can also read about the Quran angle, specifically, in the fallout over a discovery of an ancient Quran manuscript last summer.

(2) Why did the Reformation split Europe, and more specifically the Holy Roman Empire. In 1515, Germans chased indulgences and venerated Eucharist wafers that bled. In 1525, some Germans chased indulgences and venerated wafers; and some Germans burned them. Why? This debate has a LONG way to go, with scholars still arguing over whether the Reformation was "really" an intellectual event (MacCulloch), a social one (e.g. Blickle), a political one (Brady), merely one manifestation of longer term shifts (Wallace). One really interesting development is William Bradford Smith's reconstruction of towns versus countryside in 15th-16th century Franconia, where he shows how an urban patriciate-rural nobility split began to develop in the late Middle Ages that would coalesce as religious difference after the Reformation. He still can't explain the spiritual shift, though, which suggests that some combination of pragmatic/realpolitik and intellectual-spiritual factors might be in play.

(3) Where did the Western ideas of race and racism come from? How do we get from xenophobia and religious prejudice to "biological" racism? This is the topic that so many medievalists want to dip a toe in at the end of their major book, from Robert Bartlett (who sees internal and external colonization in the expansion of western Europe to the east in 1000-1300 as establishing legal differentiation; *The Making of Europe) to David Nirenberg (arguing that it's in late medieval Spain that anti-Judaism becomes anti-Semitism in the eyes of the Inquisition: the descendants of Jewish converts to Christianity are suspected of being Jews because there is some biological component to what was previously "just" a set of beliefs and practices).

(4) What was the impact of the Black Death, especially on the world system? The initial epidemic and its early successive waves massacred Asia, the Near East, Eastern and Western Europe. How did the patterns of demographic devastation in conjunction with the intellectual reactions (emerging out of different cultural-religious-intellectual contexts) affect "Western" versus "Eastern" civilization? Did the Black Death really cause anything, or did it accelerate or decelerate movements already in progress?

(5) How much, really, did it suck to live in the late Middle Ages?

History as a discipline doesn't generally coalesce around "unsolved problems" in the same way that science and medicine often does. There are interesting reasons for this — it has been argued that part of the definition of a "science" is that many practitioners all acknowledge they are trying to solve very similar problems, which relies on a huge body of shared interest and accepted ways of viewing the world. Historians are a funny bunch in that they are empirical (they rely on facts and evidence) but they sit somewhere between the social sciences (with their generally more rigid methodologies) and the more creative humanities (with their aesthetics and general rejection of methodology).

What historians tend to coalesce around are broad themes and lenses of doing history, not unanswered questions. You could rephrase these as "questions," I suppose, but they are unlikely to have "answers" in the sense you probably mean.

So a big theme for awhile has been transnationalism, which has been moving through various different sub-disciplines of history for a few decades now. To understand transnationalism you have to understand what it is pitting itself against: nationalist histories (focused on a single nation), internationalist histories (the same framework as nationalist histories but applied to a few more nations), and globalization (which is more about the process of nations thinking more globally, and arguably implies a sort of teleology or sense of "progress"). The transnational approach was to focus on the idea of nations and even the idea of "the international" as something in flux, developed out of the course of said history, rather than the preconditions for it.

Another one, from the discipline of science and technology studies (which includes history of science, my discipline) is the idea of coproduction. Again, to understand it, you have to understand what it is pitched against. For a long time (as in, is goes back to 19th century thinkers) there were questions asked about whether science and technology determined changes in society, or whether society was responsible for changes in science and technology. That is, who affects who? And of course you can find examples of each case, and, of course, various people have tried various forms of a synthetic argument to take into account both actions (e.g. Thomas Hughes argued for "technological momentum," which basically means that society starts out "in control" but eventually the accretion of time leads to a logic imposed by the technology). Coproduction is a different sort of solution to this kind of question: instead of asking which entity affects which entity, it says that both of these entities are in flux (that is, there is no solid thing called "society" or "science"), and that they coproduce, co-create, one another simultaneously. To put it another way, statements about nature are statements about society and vice versa. Instead of asking, did Darwin come up with the theory of evolution because of his social influences, or did Victorian England modify itself based on the theory of evolution, you say, well, these things are actually pretty well bound up with one another, and the theory of evolution became a way to talk about what kinds of entities science was and what kinds of entities society was. That is, in the discussions people had about it (including Darwin's), they are defining, either explicitly or implicitly, what counts as "nature" and what counts as "society," and these things are not "givens" to be analyzed about who impacts who.

OK, you're saying, jeez louise, this is not what I was asking! What I was asking was, are there any cool mysteries out there that scholars are trying to solve, not "what are the latest jargony fads in academic history"! And hey, I'm with you on that. I find the above stuff sometimes useful as a way to think about what I'm studying (I am on a panel about transnationalism in Cold War technology this fall, and I do find coproduction an interesting idea), but you have to be pretty well signed on to the academic lifestyle to think these are fun mysteries.

Individual historians often are trying to solve unanswered questions in their specific research areas. I know I am! One paper I've been trying to get out the door is a very micro-historical attempt to understand what evidence we have for how Harry Truman felt about the atomic bombs prior to the surrender of Japan. (That is, between him learning about them after FDR's death, but most importantly the period in which he knew they were going to be used, was initially elated by the first one's success, and then becomes much more morose about the whole thing once casualty figures come in.) There are loads of interesting questions in this little slice of history. Did Truman know what kind of target Hiroshima was? (That is, a city with a base in it, not just a base.) Did he know a second bomb would be used so soon? Who wrote his speeches in which he characterized Hiroshima as a base and not a city? When, exactly, did he learn that thousands of women and children were killed in the attacks? Can we trace the evolution of his thoughts reliably through direct statements, diary entries of his contemporaries, and maybe even a little linguistic analysis? (Truman, like many people, tends to have "stock phrases," so you can see where certain things repeat.) Why exactly did Truman order no further atomic bombings on August 10th — was it because he felt they would hamper negotiations, or because he was put off by the slaughter of civilians, or because he felt like the bombings were being done out of his direct control? What do the atomic bombings and Truman and Stimson's response to them tell us about the evolution of the idea that the use of atomic bomb was a "civilian" responsibility as opposed to a "military" one? When exactly did Truman's public justification of what happened turn into the "saving lives" one?

And so on and so on. You can see from the above that once you corner any individual historian (or, at least, me) and start quizzing them on what questions are on their mind, you'll get a million of them. Those are the questions that interest me at the moment. They probably interest a few other historians though very few are working on those particular questions right now. In fact many historians deliberately choose topics where other people aren't working on the same questions because they don't like to feel like they are in "competition" with one another. It is considered a bad thing to work on exactly the same book or article topic, for example. Junior historians are deathly afraid of being scooped — because how many historians do you need puzzling over whether Truman knew there were two bombs?

All of which is a very unsatisfying answer, I am sure, but you can see why I prefaced it with a discussion of why historians aren't scientists. (And thank goodness, from my perspective! If I wanted to be hidebound to a methodology I'd have become a sociologist or something. Smiley face and wink to my dear sociologist friends.) These aren't questions on the level of "what is dark energy?" or "can you make general relativity compatible with quantum mechanics?" Arguably those aren't the questions that most practicing physicists are working on, either — though their day-to-day- micro-questions (e.g., "why does star X rotate at the speed it does?") might be more deliberately tied into those "big" questions of their discipline than a historian's questions might be tied into theirs.

One major, final difference to contemplate is whether historians are ever satisfied with the answers to their questions. This, too, is a major disciplinary difference and arguably a definition for what a "science" is. You can always, always have doubts about an answer to a question, even in a "hard" science like physics or chemistry. But it is characteristic of scientific communities that they tend to jettison those doubts after a reasonably short amount of time. If they get a pretty good answer, they say, great — on to the next questions. This is why sciences appear to make "progress," moving through new questions at a quick pace. (Historians of science have whole reams of dead forests dedicated to the tricky notion of "progress" in science but let's put that aside for now.) History doesn't really work that way. You do get programs or answers or lenses that hold sway for awhile. But a generation or two later, the later historians do their best to explain that the reigning approach is probably wrong, and put forward a new one. And there are lots of possible reasons for this (ranging from the cynical to the idealistic — are they just finding a way to say new things about old topics, or is it a reflection of genuinely better understanding over time?), but it remains a characteristic of the discipline, that it repeatedly "kills its fathers." To many in the discipline I think this represents a form of "progress," though I think the historians of science (who are picky about these kinds of things) might throw some cold water on that issue.

I hope the above is interesting even if it is not exactly what you asked for (but hey, it's how a tricky historian of science would approach the answer). If you're interested in notions of collective progress in the development of knowledge, Thomas Kuhn's The Structure of Scientific Revolutions (1962) is the classic text in the history of science, though of course most historians of science (myself included) don't consider it to be entirely correct (Kuhn is a father who we've long since killed but still keep around because he makes for a good introduction to these topics).