  We study the ground-state and low-energy properties of classical vector spin
models with nearest-neighbour antiferromagnetic interactions on a class of
geometrically frustrated lattices which includes the kagome and pyrochlore
lattices. We explore the behaviour of these magnets that results from their
large ground-state degeneracies, emphasising universal features and systematic
differences between individual models. We investigate the circumstances under
which thermal fluctuations select a particular subset of the ground states, and
find that this happens only for the models with the smallest ground-state
degeneracies. For the pyrochlore magnets, we give an explicit construction of
all ground states, and show that they are not separated by internal energy
barriers. We study the precessional spin dynamics of the Heisenberg pyrochlore
antiferromagnet. There is no freezing transition or selection of preferred
states. Instead, the relaxation time at low temperature, T, is of order
hbar/(k_B T). We argue that this behaviour can also be expected in some other
systems, including the Heisenberg model for the compound SrCr_8Ga_4O_{19}.
