  Let $G$ be a permutation group on a finite set $\Omega$. A subset $B
\subseteq \Omega$ is a base for $G$ if the pointwise stabilizer of $B$ in $G$
is trivial. The base size of $G$, denoted $b(G)$, is the smallest size of a
base. A well known conjecture of Pyber from the early 1990s asserts that there
exists an absolute constant $c$ such that $b(G) \le c\log |G| / \log n$ for any
primitive permutation group $G$ of degree $n$. Some special cases have been
verified in recent years, including the almost simple and diagonal cases. In
this paper, we prove Pyber's conjecture for all non-affine primitive groups.
