To any Riemannian manifold of dimension n is
associated a closed subgroup of SO(n), the holonomy group -- a basic
invariant in Riemannian geometry. A famous theorem of Berger gives a complete (and rather
small) list of the groups which can appear. Surprisingly, the compact manifolds
with holonomy smaller than SO(n) are all related in some way to
algebraic manifolds. I will try to explain how this works, and how this leads to
interesting problems in algebraic geometry.