Let G be a group acting (faithfully) on a set X. A base for this action is a subset Y of X so that no element of G fixes every element of Y. The question of what is the minimal size of a base is a classical subject going back to the early days of finite permutation group theory. In this talk I will mostly focus on the case that G is simple algebraic group and X is an irreducible variety. A closely related problem is to determine a generic stabilizer (if it exists). Note that base size 1 is the same as saying some stabilizer is trivial (and indeed base size b on X is the same as saying base size 1 on b copies of X). We will consider the case where X = G/H for some maximal closed subgroup H and for the case that X is an irreducible rational G-module. Even if one is only interested in the case of finite groups, these cases are relevant. Some of this is joint work with Burness and Saxl, some with Lawther and some with Garibaldi. Please note: There will be a pre-talk for graduate students from 2:30 - 3:00. The regular talk will begin at 3:00.