  There are compelling reasons to believe that the dark matter of the universe
is constituted, in large part, by non-baryonic collisionless particles with
very small primordial velocity dispersion. Such particles are called cold dark
matter (CDM). The leading candidates are axions and weakly interacting massive
particles (WIMPs). The word ``collisionless'' indicates that the particles are
so weakly interacting that they move purely under the influence of gravity.
Galaxies are surrounded by CDM and hence, because of gravity, CDM keeps falling
onto galaxies from all directions. CDM infall produces a discrete number of
flows and caustics in the halos of galaxies. There are two types of caustics in
the halos of galaxies: inner and outer. An outer caustic is a simple fold
catastrophe located on a topological sphere enveloping the galaxy. An inner
caustic is a closed tube whose cross-section is an elliptic umbilic
catastrophe, with three cusps. In space, caustics are the boundaries that
separate regions with differing numbers of flows. One side of a caustic surface
has two more flows than the other. The density of CDM particles becomes very
large as one approaches the caustic from the side with the extra flows. Dark
matter caustics have specific density profiles and therefore precisely
calculable gravitational lensing properties. This dissertation presents a
formalism that simplifies the relevant calculations, and apply it to four
specific cases.
