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.