We study classical and quantum Heisenberg antiferromagnets with exchange
anisotropy of XXZ--type and crystal field single--ion terms of quadratic
and cubic form in a field. The magnets display a variety of phases,
including the spin--flop (or, in the quantum case, spin--liquid) and
biconical (corresponding, in the quantum lattice gas description,
to supersolid) phases. Applying ground--state considerations, Monte
Carlo and density matrix renormalization group methods, the
impact of quantum effects and lattice dimension is analysed. Interesting
critical and multicritical behaviour may occur at quantum and
thermal phase transitions.