We will present a self-contained introduction to the Castelnuovo–Mumford regularity for standard graded rings over general Noetherian base rings. In particular, we will show that it can be defined in terms of the vanishing of local cohomology modules, vanishing of Koszul homology and the shifts in a minimal free resolution. We will present a proof of a classical result on regularity of the powers of an ideal due originally to Cutkosky, Herzog and Trung and, independently, to Kodiyalam. Variants for products of powers and generalizations will be discussed as well. Examples of determinantal or combinatorial nature will be presented.