  Spin systems with hyperbolic symmetry originated as simplified models for the
Anderson metal--insulator transition, and were subsequently found to exactly
describe probabilistic models of linearly reinforced walks and random forests.
In this survey we introduce these models, discuss their origins and main
features, some existing tools available for their study, recent probabilistic
results, and relations to other well-studied probabilistic models. Along the
way we discuss some of the (many) open questions that remain.
