  We study the vortex lines that are a feature of many random or disordered
three-dimensional systems. These show universal statistical properties on long
length scales, and geometrical phase transitions analogous to percolation
transitions but in distinct universality classes. The field theories for these
problems have not previously been identified, so that while many numerical
studies have been performed, a framework for interpreting the results has been
lacking. We provide such a framework with mappings to simple supersymmetric
models. Our main focus is on vortices in short-range correlated complex fields,
which show a geometrical phase transition that we argue is described by the
CP^{k|k} model (essentially the CP^{n-1} model in the replica limit
n\rightarrow 1). This can be seen by mapping a lattice version of the problem
to a lattice gauge theory. A related field theory with a noncompact gauge
field, the 'NCCP^{k|k} model', is a supersymmetric extension of the standard
dual theory for the XY transition, and we show that XY duality gives another
way to understand the appearance of field theories of this type. The
supersymmetric descriptions yield results relevant, for example, to vortices in
the XY model and in superfluids, to optical vortices, and to certain models of
cosmic strings. A distinct but related field theory, the RP^{2l|2l} model (or
the RP^{n-1} model in the limit n\rightarrow 1) describes the unoriented
vortices which occur for instance in nematic liquid crystals. Finally, we show
that in two dimensions, a lattice gauge theory analogous to that discussed in
three dimensions gives a simple way to see the known relation between
two-dimensional percolation and the CP^{k|k} sigma model with a \theta-term.
