Lectures presented at the summer course on ‘‘Low Dimensional Quantum Field Theories for Condensed Matter Physicists’’,

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 (i) collinear spin correlations, spin-Peierls or valence-bond-solid order, and confined spinons, order and confined spinons or (ii) coplanar spin correlations, no spin-Peierls order and deconfined bosonic spinons. 1