  Let $G$ be a permutation group on a finite set $\Omega$. The base size of $G$
is the minimal size of a subset of $\Omega$ with trivial pointwise stabiliser
in $G$. In this paper, we extend earlier work of Fawcett by determining the
precise base size of every finite primitive permutation group of diagonal type.
In particular, this is the first family of primitive groups arising in the
O'Nan-Scott theorem for which the exact base size has been computed in all
cases. Our methods also allow us to determine all the primitive groups of
diagonal type with a unique regular suborbit.
