  Lens models appropriate for representing cusped galaxies and clusters are
developed. The analogue of the odd number theorem for cusped density
distributions is given. Density cusps are classified into strong, isothermal or
weak, according to their lensing properties. Strong cusps cause multiple
imaging for any source position, whereas isothermal and weak cusps give rise to
only one image for distant sources. Isothermal cusps always possess a
pseudo-caustic. When the source crosses the pseudo-caustic, the number of
images changes by unity.Two families of cusped galaxy and cluster models are
examined in detail. The double power-law family has an inner cusp, followed by
a transition region and an outer envelope. One member of this family -- the
isothermal double power-law model -- possesses an exceedingly scarce property,
namely, the lens equation is exactly solvable for any source position. This
means that the magnifications, the time delay and the lensing cross-sections
are all readily available. The model has a three dimensional density that is
cusped like $r^{-2}$ at small radii and falls off like $r^{-4}$ asymptotically.
Thus, it provides a very useful representation of the lensing properties of a
galaxy or cluster of finite total mass with a flat rotation curve. The second
set of models studied is the single power-law family. These are single density
cusps of infinite extent. The properties of the critical curves and caustics
and the behaviour of the lenses in the presence of external shear are all
discussed in some detail.
