  A spin network is a generalization of a knot or link: a graph embedded in
space, with edges labelled by representations of a Lie group, and vertices
labelled by intertwining operators. Such objects play an important role in
3-dimensional topological quantum field theory, functional integration on the
space A/G of connections modulo gauge transformations, and the loop
representation of quantum gravity. Here, after an introduction to the basic
ideas of nonperturbative canonical quantum gravity, we review a rigorous
approach to functional integration on A/G in which L^2(A/G) is spanned by
states labelled by spin networks. Then we explain the `new variables' for
general relativity in 4-dimensional spacetime and describe how canonical
quantization of gravity in this formalism leads to interesting applications of
these spin network states.
