  We study the complex Monge-Amp\`ere operator in bounded hyperconvex domains
of $\C^n$. We introduce a scale of classes of weakly singular plurisubharmonic
functions : these are functions of finite weighted Monge-Amp\`ere energy. They
generalize the classes introduced by U.Cegrell, and give a stratification of
the space of (almost) all unbounded plurisubharmonic functions. We give an
interpretation of these classes in terms of the speed of decreasing of the
Monge-Amp\`ere capacity of sublevel sets and solve associated complex
Monge-Amp\`ere equations.
