Africanists tend to rely on matrices of oral history for two [main types of] cases.

First, if a society is nonliterate (which is different from "illiteracy," in that it's not a case of lack but a case of distinction), we look for epics, usually poetic or performed, that have come across time and name people as well as events and locales. Nonliterate societies tended to have mechanisms for transferring this knowledge that were fairly robust: unlike the telephone game we all probably played in grade school, this was an entire caste moving knowledge from generation to generation. It would still change, especially in response to the needs of the present for the knowledge, but those additions and idiosyncrasies can be found and often eclecticism was limited by it. The signal example of this is the story of Sundiata Keita, the first great mansa of a united Mali. A number of transliterations exist including one in film, each marked by the circumstances of its collection and its purpose, but virtually all run through the Kouyaté clan of djeli or griots. Even then, most of the oral history deals with the political elite and the privileged; sometimes you'll get at important caste histories in that way too (for ironworkers, perhaps). But the majority of people mostly remain left out, much as they do with written historical records.

So in cases where a society has been connected to literacy, oral history is also a way of getting at the recent past for the many people left out of the written record. Sometimes this operates as the collection of oral testimony in the field, and other times as family histories or tales that elucidate something left unelaborated in a written record. Subaltern studies deals with a lot of this kind of knowledge and the pitfalls of ever fairly representing the speaker, and it's not at all unique to Africa. But it can add a tremendous amount of knowledge about the context of state documents, which have their own positional issues.

In both cases, the strength of the oral evidence is greatly enhanced with corroboration by written documents, archaeological data, or other kinds of additional support. But that's true of almost any historical matter, even those in writing.

When studying the history of populations without a written historical record we rely on a multidisciplinary approach that combines information from a variety of fields. The most common sources of data for my specific area of focus, the spread of disease in protohistoric North America, arise from several fields of study including ethnohistory and oral tradition, ecology, archaeology, and bioarchaeology. I'll provide a couple of brief examples of how these areas of research inform my studies into the impact and timing of disease in North America.

Ethnohistory and oral tradition provide insight into past cultures, as well as the story of extant cultures, as passed from record keeper to record keeper with each generation. While our modern Western dependence on written sources might make us think oral tradition is inherently unreliable, for the bulk of human history we relied on the oral transmission of knowledge from person to person, generation after generation, for hundreds of thousands of years. Cultures with rich oral tradition place tremendous importance on the correct presentation, transmission, and re-telling of this knowledge to prevent errors in repetition. One of my favorite examples of an oral tradition is the Plains Winter Counts.

Northern Plains tribes (like the Lakota, Kiowa, Mandan, and Dakota) kept historical records in the form of Winter Counts. Winter Counts were a historical record, a list of year names representing the most significant events in the life of the band. Pictorial representations of that event served as a reminder, a kind of mnemonic device, for the keeper of the count to retell the history of the band. We know of 53 Winter Counts that together provide a historical record of the Northern Plains from 1682 to 1920. By compiling the Winter Counts together into a master narrative we can establish a chronology, cross-check errors, and be fairly certain the events depicted are accurate to roughly two years. From this narrative we can determine the frequency and impact of infectious disease on the Northern Plains populations before the arrival of permanent European-descent settlers. You can read more from a previous answer I gave here, and be sure to check out the Smithsonian's Online Winter Count Exhibit if you would like to know more.

Briefly, the Winter Counts show us epidemics of infectious disease arrived before significant, sustained face-to-face contact with Europeans (3-5 epidemics before the establishment of permanent trading posts). Epidemics of infectious disease arrived in waves, one roughly every 5 to 10 years, burned through the pool of susceptible hosts, and left long periods of stasis in their wake. An entire generation could be born, live and die between waves of disease for some bands, while others were hit with multiple events in quick succession. Even in the same epidemic of the same pathogen, mortality could differ based on immunity from previous exposure and the stressors (famine, poor nutrition, displacement, etc.) influencing the health of the band. Winter Counts tell a story that might be lost without an oral tradition, and provide a window into understanding life on the northern Plains in the protohistoric.

Second, we rely on studies in ecology to help us understand how the human modification of the environment changed over time. One of my favorite examples just came out recently, and looked at how the cyclic fire patterns of forest fires in a specific region of New Mexico changed over time. The question posed by the research was did excess mortality due to disease strike early in the contact period, coinciding with early entradas, or did disease arrive later with sustained contact during the mission period, or was there population stasis throughout the entire period. Liebman and colleagues found evidence of population stasis from 1492-1620, with drastic decline at Jemez in the following six decades, coinciding with the establishment of the New Mexico missions. As population changed so did the environment, with evidence of reforestation as the people living around Jemez constricted in the wake of disease, warfare, and famine. The ecological story helps us understand what was happening beyond the sight of the births, deaths, and baptisms recorded in the spotty mission ledgers in the nearby pueblos. We see the larger environment change in response to population loss, and combined when combined with oral history and the written history of the period a fuller picture of the excess mortality of contact emerges.

Finally, archaeology and bioarchaeology allow us another window into the past. In the early and mid 20th century many scholars believed epidemic diseases spread universally, and catastrophically, ahead of European explorers, transforming the New World before their true, untouched state could be detailed in the historical record. In the past several decades the tremendous work of both archaeologists and historians in the Southeast U.S. revealed a more nuanced story, and allowed us to place the patterns observed during the protohistoric into the longer history of the region. Archaeological investigations showed a continuity of site use, and a pattern of chiefdom development and decline persisted, as it had for centuries, for much of the protohistoric period. Along the Gulf Coast, even at sites directly in the path of de Soto's entrada who endured sustained contact for five months when the conquistador wintered nearby, like the Governor Martin site, the archaeological evidence does not show catastrophic decline like we might expect if early, universal disease winnowed the population. Like in New Mexico, the impact of disease as evidenced by disruption of previous archaeological patterns indicating population decline, seems to appear later, with the establishment of missions, nearly a century after the initial Spanish forays into the Southeast. Similarly, cemetery samples from the same period show an ordered, consistent deposition of the departed, similar to earlier patterns, and not the dramatic spikes in mortality one might expect with disease spread. Hutchinson's landmark study of Tatham Mound showed that even in an early cemetery with individuals displaying markers of interpersonal violence from Spanish weapons, there was no evidence of catastrophic mortality from epidemic disease.

I hope these few brief examples show how I use other sources beyond the written word to research a fascinating period in human history, and hopefully understand one of the major questions in the field. Taken together, ethnohistory, oral tradition, ecology, archaeology, and bioarchaeology help to deepen our understanding of times and places beyond the view of historical sources, and give voice to people who might otherwise be ignored by the pages of history.

Archaeology is a great resource where written language is scarce or missing. A great example is the Indus Valley Civilization in India and Pakistan.

We still have not been able to descipher the script of the Indus Valley. The script appears on a number of seals but defies interpretation. So whatever information we have has been gleaned from archaeological remains.

The Indus Valley Civilization is famous for its town-planing. The two best examples are Harappa and Mohenjodaro. The towns are planned in a grid system with evenly spaced roads of regular width. Wells and dustbins are present at intervals. Houses are very well planned with large rooms and bathrooms. There is extensive evidence of a developed underground sewage system.

What all of this indicates is the presence of an economically powerful central authority which planned and implemented this urban structure. Again the urban structure also shows indications of class difference. Apart from the large well planned houses we have rows of one/two room tenements at the peripheries of the towns. In Mohenjodaro there is central raised citadel type area with very large structures. Thus a three tier class system of ruling-middle-lower has been theorized.

Another great resource are the manufactured goods - pottery (most common) jewellery, seals, trade goods, tools etc. Indus Valley had a very distinctive bead based jewellery using Lapis Lazuli and other materials found locally as well as pretty far away in South India. This indicates a good local trade network. But Indus Valley goods have been found all over the ancient world, as far away as Mesopotamia which indicates a strong international trade network.

These are only some of the many areas of information gleaned from archaeology. I would strongly advice you to check out the works of J.M. Kenoyer and Raymond Allchin. In addition to analysis the books have great illustrations of the stunningly beautiful ceramics and jewellery.

I can tangentially answer this question, as I am currently studying Carthage. After the fall of Carthage, we have no surviving pieces from Carthagian authors- only from Roman and Greek sources. We piece together evidence based on a few sources- archaeologic (pottery, bones, housing ruins); paraphrased sources (books that say stuff like "As you will recall from Mago's 75th doctrine, ...") and foreign historians. Each of these have their problems. Archaeologic sources are difficult to interpret, and opinions of their subject matter often differ. Paraphrased sources don't list the entire work, because they are just citing a source, not rewriting it (much like we do at askhistorians!). Foreign historians are often heavily biased.

As an example, the tophet in Carthage is a complicated subject, because we aren't sure what it is and why it happened. Archaeological evidence is often contradictory, and two papers using the same source can have a different opinion. The paraphrased sources use different and confusing terms. Finally, all of the foreign historians are heavily biased, because, ya know, they fought the Carthagians for a century.