  The quantum dynamics of mesoscopic or macroscopic systems is always
complicated by their coupling to many "environmental" modes.At low T these
environmental effects are dominated by localised modes, such as nuclear and
paramagnetic spins, and defects (which also dominate the entropy and specific
heat). This environment, at low energies, maps onto a "spin bath" model. This
contrasts with "oscillator bath" models (originated by Feynman and Vernon)
which describe {\it delocalised} environmental modes such as electrons,
phonons, photons, magnons, etc. One cannot in general map a spin bath to an
oscillator bath (or vice-versa); they constitute distinct "universality
classes" of quantum environment. We show how the mapping to spin bath models is
made, and then discuss several examples in detail, including moving particles,
magnetic solitons, nanomagnets, and SQUIDs, coupled to nuclear and paramagnetic
spin environments. We show how to average over spin bath modes, using an
operator instanton technique, to find the system dynamics, and give analytic
results for the correlation functions, under various conditions. We then
describe the application of this theory to magnetic and superconducting
systems.Particular attention is given to recent work on tunneling magnetic
macromolecules, where the role of the nuclear spin bath in controlling the
tunneling is very clear; we also discuss other magnetic systems in the quantum
regime, and the influence of nuclear and paramagnetic spins on flux dynamics in
SQUIDs.
