If G is a group of permutations on a set X, then a base for this action is a subset so that no nontrivial element fixes all the elements in the subset. This has been studied for well over a century. It has applications in computational group theory. Note if |X| = n and there is a base of size b, then |G| is at most b^n. Similarly, if dim X = d and there is a base of size b, then dim G is at most bd. In joint work with Burness and Saxl, we are working on a program to classify the sizes of the smallest base for simple groups acting primitively on a set (both for finite and algebraic groups). In this talk, I will focus on the case of simple algebraic groups where we have an almost complete classification. One of the key ideas is to focus on the case when almost all b-tuples form a base. There are some interesting examples where there is a base of size b but the generic stabilizer of a set of size b is finite but nontrivial.