  We construct a series of 2+1-dimensional models whose quasiparticles obey
non-Abelian statistics. The adiabatic transport of quasiparticles is described
by using a correspondence between the braid matrix of the particles and the
scattering matrix of 1+1-dimensional field theories. We discuss in depth
lattice and continuum models whose braiding is that of SO(3) Chern-Simons gauge
theory, including the simplest type of non-Abelian statistics, involving just
one type of quasiparticle. The ground-state wave function of an SO(3) model is
related to a loop description of the classical two-dimensional Potts model. We
discuss the transition from a topological phase to a conventionally-ordered
phase, showing in some cases there is a quantum critical point.
