The authors address the classical problem of determining finite primitive permutation groups $G$ with a regular subgroup $B$. The main theorem solves the problem completely under the assumption that $G$ is almost simple. While there are many examples of regular subgroups of small degrees, the list is rather short (just four infinite families) if the degree is assumed to be large enough, for example at least 30!. Another result determines all primitive groups having a regular subgroup which is almost simple. This has an application to the theory of Cayley graphs of simple groups.