  These are lectures presented at the summer course on ``Low Dimensional
Quantum Field Theories for Condensed Matter Physicists'', 24 Aug. to 4 Sep.
1992, Trieste, Italy. I review recent work, performed in collaboration
primarily with N. Read and Jinwu Ye, on the properties of quantum
antiferromagnets in two dimensions. The emphasis is on the properties of the
antiferromagnet in states which do not have any long-range magnetic order. The
universal spin dynamics in the quantum critical region of number of frustrated
and random antiferromagnets is studied; implications for neutron scattering
experiments in the lightly-doped cuprates are noted. The nature of the
quantum-disordered phase of non-random frustrated antiferromagnets is examined
in some detail: the states found have ({\em i\/}) collinear spin correlations,
spin-Peierls or valence-bond-solid order, and confined spinons, order and
confined spinons or ({\em ii\/}) coplanar spin correlations, no spin-Peierls
order and deconfined bosonic spinons.
