  This article is concerned with perfect isometries between blocks of finite
groups. Generalizing a method of Enguehard to show that any two p-blocks of
(possibly different) symmetric groups with the same weight are perfectly
isometric, we prove analogues of this result for p-blocks of alternating groups
(where the blocks must also have the same sign when p is odd), of double covers
of alternating and symmetric groups (for p odd, and where we obtain crossover
isometries when the blocks have opposite signs),of complex reflection groups
G(d,1,n) (for d prime to p), of Weyl groups of type B and D (for p odd), and of
certain wreath products. In order to do this, we need to generalize the theory
of blocks, in a way which should be of independent interest.
