I will give an overview of recent work on the physics of lattice models of
close-packed loops. The models have two phases -- one in which all loops
are finite, and another in which some loops are extended -- and the
interest is in universal features of these phases and the transitions
between them. The models describe a number of problems in classical
statistical physics -- where, for example, the loops may be vortex
lines in a three-dimensional random field -- and also many-body quantum
systems -- where the loops are world-lines of particles.