  As Charles Goodyear discovered in 1839, when he first vulcanised rubber, a
macromolecular liquid is transformed into a solid when a sufficient density of
permanent crosslinks is introduced at random. At this continuous equi- librium
phase transition, the liquid state, in which all macromolecules are
delocalised, is transformed into a solid state, in which a nonzero fraction of
macromolecules have spontaneously become localised. This solid state is a most
unusual one: localisation occurs about mean positions that are distributed
homogeneously and randomly, and to an extent that varies randomly from monomer
to monomer. Thus, the solid state emerging at the vulcanisation transition is
an equilibrium amorphous solid state: it is properly viewed as a solid state
that bears the same relationship to the liquid and crystalline states as the
spin glass state of certain magnetic systems bears to the paramagnetic and
ferromagnetic states, in the sense that, like the spin glass state, it is
diagnosed by a subtle order parameter.
  In this review we give a detailed exposition of a theoretical approach to the
physical properties of systems of randomly, permanently crosslinked
macromolecules. Our primary focus is on the equilibrium properties of such
systems, especially in the regime of Goodyear's vulcanisation transition.
