  We consider a general class of (intersecting) loop models in D dimensions,
including those related to high-temperature expansions of well-known spin
models. We find that the loop models exhibit some interesting features - often
in the ``unphysical'' region of parameter space where all connection with the
original spin Hamiltonian is apparently lost. For a particular n=2, D=2 model,
we establish the existence of a phase transition, possibly associated with
divergent loops. However, for n >> 1 and arbitrary D there is no phase
transition marked by the appearance of large loops. Furthermore, at least for
D=2 (and n large) we find a phase transition characterised by broken
translational symmetry.
