An acquaintance of mine just wrote an article called the Myth of Caliphate: the political history of an idea in Foreign Affairs (he's still in grad school but is going to be a bomb public intellectual soon--his blog Afternoon Map is tremendously popular among Middle East historians and he's been picked up by many other places, including I think Slate and the Atlantic). This article is mostly about modern attempts to revive the Caliphate after Ataturk's abolishment of the office, however, rather than its pre-Modern, pre-Ottoman history. It's far from exhaustive, aimed more at a policy audience than an academic one, but it may be a good start for thinking about current issues (if that's what inspired your question). I think one key line of his is:

The caliphate’s more recent history under the Ottomans shows why the institution might be better thought of as a political fantasy—a blank slate just as nebulous as the “dictatorship of the proletariat”—that contemporary Islamists are largely making up as they go along.

I wouldn't quite go that far, but I think it's fair to say as a modern political idea it is not slavishly tied to historical models.

The ending is rad:

By treating the Ottoman caliphate as the final historical reference point for what current Islamists aspire to, Western pundits conflate the contemporary dream of a powerful, universally respected Muslim leader with the late Ottoman sultan's failed dream of becoming such a figure himself. The circumstances uniting these dreams—and the appeal of strong religious power in the face of Western political, military, and economic power—may be the same. But so are the challenges. Contemporary claimants to the title of caliph may quickly find themselves in the same boat as Ottoman caliphs. Political or military success, rather than history or theology, can bring short-lived legitimacy, but failure in these realms will bring other contenders for power.

Edit: it's worth noting that there have been several different ways the next Caliph was selected over history. The first four by Sunni reckoning were selected by shura, meaning "council" or "consultation", though after that point it became dynastic and one of the Caliph's sons (or other close male relative) would inherit the caliphate. Often this was eldest son, but in other cases, notably with the early Ottomans, there would be brief civil war of dynastic succession. Sunni Caliphates are mainly (but not exclusively) declared by who controls Mecca and Medina, or sometimes people who used to control Mecca but were expelled by a rival Caliph claimed that they were the rightful rulers of Mecca and Medina even if they didn't actually control Mecca and Medina now. For instance, the Ottomans only declared themselves Caliphs after they conquered Mecca and Medina, just like (as Nick Danforth points out above) they declared themselves Roman Emperors after they conquered Constantinople a few decades before. There were, however, some, most notably the Umayyad Caliphs of Cordoba, who called themselves caliphs with only very distant historical ties to Mecca and Medina (the first Umayyad Caliph of Cordoba was declared in 929 CE, when the Abbasids had taken control of the Caliphate from the Umayyad line almost two centuries earlier, in 750 CE).

Shi'a reckoning of Caliphs have always been about (before any theological differences, this political difference was the dividing line between Shi'a and Sunni), though there have been a lot of different lines--Sevener, Twelver, and several lines, such as the Agha Khanis and Bohra, who are still ruled by an unbroken line of descent from Ali, Fatimah, and Muhammed. After the Fatimid Caliphs were Sevener Shi'as who, once the line of descent from Ali was broken, I believe argued that they were caliphs rather than imams.

It's worth noting that many Shi'a Caliphs and some Sunni Caliphs weren't necessarily powerful temporal rulers.

To piggyback off this question, were there any non-Islamic caliphates?