  We explicitly determine all of the transitive groups of degree p-squared, p a
prime, whose Sylow p-subgroup is not the wreath product of two cyclic groups of
order p. Furthermore, we provide a general description of the transitive groups
of degree p-squared whose Sylow p-subgroup is such a wreath product, and
explicitly determine most of them. As applications, we solve the Cayley
Isomorphism problem for Cayley objects of an abelian group of order p-squared,
explicitly determine the full automorphism group of Cayley graphs of abelian
groups of order p-squared, and find all nonnormal Cayley graphs of order
p-squared.
