  Dimensionally-regulated Feynman integrals are a cornerstone of all
perturbative computations in quantum field theory. They are known to exhibit a
rich mathematical structure, which has led to the development of powerful new
techniques for their computation. We review some of the most recent advances in
our understanding of the analytic structure of multiloop Feynman integrals in
dimensional regularisation. In particular, we give an overview of modern
approaches to computing Feynman integrals using differential equations, and we
discuss some of the properties of the functions that appear in the solutions.
We then review how dimensional regularisation has a natural mathematical
interpretation in terms of the theory of twisted cohomology groups, and how
many of the well-known ideas about Feynman integrals arise naturally in this
context. This is Chapter 3 of a series of review articles on scattering
amplitudes, of which Chapter 0 [arXiv:2203.13011] presents an overview and
Chapter 4 [arXiv:2203.13015] contains closely related topics.
