We show that Q-balls naturally exist in the Minimal Supersymmetric Standard Model (MSSM) with soft SUSY breaking terms of the minimal N = 1 SUGRA type. These are associated with the F-and D-flat directions of the scalar potential once radiative corrections are taken into account. We consider two distinct cases, corresponding to the "HuL" (slepton) direction with L-balls and the "u(c)d(c)d(c)" and "u(c)u(c)d(c)e(c)" (squark) directions with B-balls. The L-ball always has a small charge, typically of the order of 1000, whilst the B-ball can have an arbitrarily large charge, which, when created cosmologically by the collapse of an unstable Affleck-Dine condensate, is likely to be greater than 10(14). The B-balls typically decay at temperatures less than that of the electroweak phase transition, leading to a novel version of Affleck-Dine baryogenesis, in which the B asymmetry comes from Q-ball decay rather than condensate decay. This mechanism can work even in the presence of additional L violating interactions or B-L conservation, which would rule out conventional Affleck-Dine baryogenesis.