  We obtain an exact formula for the average order of elements of a wreath
product of two finite groups. Then focussing our attention on $p$-groups for
primes $p$, we give an estimate for the average order of a wreath product $A\wr
B$ in terms of maximum order of elements of $A$ and average order of $B$ and an
exact formula for the distribution of orders of elements of $A\wr B.$ Finally,
we show how wreath products can be used to find several rational numbers which
are limits of average orders of a sequence of $p$-groups with cardinalities
going to infinity.
