Demographic processes directly affect patterns of genetic variation within contemporary populations as well as future generations, allowing for demographic inference from patterns of both present-day and past genetic variation. Advances in laboratory procedures, sequencing and genotyping technologies in the past decades have resulted in massive increases in high-quality genome-wide genetic data from present-day populations and allowed retrieval of genetic data from archaeological material, also known as ancient DNA. This has resulted in an explosion of work exploring past changes in population size, structure, continuity and movement. However, as genetic processes are highly stochastic, patterns of genetic variation only indirectly reflect demographic histories. As a result, past demographic processes need to be reconstructed using an inferential approach. This usually involves comparing observed patterns of variation with model expectations from theoretical population genetics. A large number of approaches have been developed based on different population genetic models that each come with assumptions about the data and underlying demography. In this article I review some of the key models and assumptions underlying the most commonly used approaches for past demographic inference and their consequences for our ability to link the inferred demographic processes to the archaeological and climate records. This article is part of the theme issue 'Cross-disciplinary approaches to prehistoric demography'.