  This note has two goals. The first is to give a short and self contained
introduction to the Castelnuovo-Mumford regularity for standard graded ring $R$
over a general base ring. The second is to present a simple and concise proof
of a classical result due to Cutkosky, Herzog and Trung and, independently, to
Kodiyalam asserting that the regularity of powers of an homogeneous ideal $I$
of $R$ is eventually a linear function in $v$. Finally we show how the
flexibility of the definition of the Castelnuovo-Mumford regularity over
general base rings can be used to give a simple characterization of the ideals
whose powers have a linear resolution in terms of the regularity of the Rees
ring.
