  We discuss the behavior of the Castelnuovo-Mumford regularity under certain
operations on ideals and modules, like products or powers. In particular, we
show that reg(IM) can be larger than reg(M)+reg(I) even when I is an ideal of
linear forms and M is a module with a linear resolution. On the other hand, we
show that any product of ideals of linear forms has a linear resolution. We
also discuss the case of polymatroidal ideals and show that any product of
determinantal ideals of a generic Hankel matrix has a linear resolution.
