  We describe random loop models and their relations to a family of quantum
spin systems on finite graphs. The family includes spin 1/2 Heisenberg models
with possibly anisotropic spin interactions and certain spin 1 models with
SU(2)-invariance. Quantum spin correlations are given by loop correlations.
Decay of correlations is proved in 2D-like graphs, and occurrence of
macroscopic loops is proved in the cubic lattice in dimensions 3 and higher. As
a consequence, a magnetic long-range order is rigorously established for the
spin 1 model, thus confirming the presence of a nematic phase.
