  We generalize a class of Heisenberg antiferromagnets in one, two and three
dimensions, which have been shown to exhibit magnetization plateaus for
spin-1/2. In a certain parameter range of the general model, which is formally
defined in D dimensions, we obtain the exact ground state(s) in the presence of
an external magnetic field for arbitrary values of spin. In this range, the
magnetization remains a constant as a function of the external field, except at
some special values of the field where there is a jump from one plateau to the
next. The plateaus are formed at certain specific fractions of the full
magnetization which are determined by the spin and the lattice. Our general
spin-S result reproduces the known cases for spin-1/2 in various lattices.
Furthermore, we argue that outside the exact regime, the mechanism for the
plateau formation is different. This results in first order phase transitions
along some of the plateaus as the coupling constant is varied. We rigorously
show the existence of such transitions for some particular cases. Finally, we
numerically analyze a spin-1 model in one dimension using exact diagonalization
to obtain its complete phase diagram. It agrees with our analytic results.
