Let G be a transitive permutation group on a set X. A base for the action is a subset B such that no nontrivial element of G acts trivially on all elements of B. The base size is the minimum cardinality of a base. We will discuss joint work with Tim Burness and Jan Saxl about computing the base size for essentially all actions of finite and algebraic groups. We will discuss connections between the algebraic group case and the finite case. We will also discuss related notions of generic base and random base size.