Let *G*&le\;Sym(&Omega\;) be a permutati
on group on a finite set &Omega\;. A base for *
G* is a subset of &Omega\; with trivial pointw
ise stabiliser\, and the base size of *G*\,
denoted *b*(*G*)\, is the minimal s
ize of a base for *G*. This classical conce
pt has been studied since the early years of permu
tation group theory in the nineteenth century\, fi
nding a wide range of applications.

Recall
that *G* is called primitive if it is tran
sitive and its point stabiliser is a maximal subgr
oup. Primitive groups can be viewed as the basic b
uilding blocks of all finite permutation groups\,
and much work has been done in recent years in bou
nding or determining the base sizes of primitive g
roups. In this talk\, I will report on recent prog
ress of this study. In particular\, I will give th
e first family of primitive groups arising in the
O'Nan-Scott theorem for which the exact base size
has been computed in all cases.