  Let $G$ be a permutation group on a set $\Omega$. A subset of $\Omega$ is a
base for $G$ if its pointwise stabilizer is trivial; the base size of $G$ is
the minimal cardinality of a base. In this paper we initiate the study of bases
for algebraic groups defined over an algebraically closed field. In particular,
we calculate the base size for all primitive actions of simple algebraic
groups, obtaining the precise value in almost all cases. We also introduce and
study two new base measures, which arise naturally in this setting. We give an
application concerning the essential dimension of simple algebraic groups, and
we establish several new results on base sizes for the corresponding finite
groups of Lie type. The latter results are an important contribution to the
classical study of bases for finite primitive permutation groups. We also
indicate some connections with generic stabilizers for representations of
simple algebraic groups.
