  We present a rigorous, detailed study of the generic, quantitative properties
of gravitational microlensing near cusp catastrophes. We derive explicit
formulas for the total magnification and centroid of the images created for
sources outside, on, and inside the cusped caustic. We obtain results on how
the image magnifications scale with respect to separation from the cusped
caustic for arbitrary source positions. Along the axis of symmetry of the cusp,
the total magnification $\mu$ scales as $\mu \propto u^{-1}$, where $u$ is the
distance of the source from the cusp, whereas perpendicular to this axis, $\mu
\propto u^{-2/3}$. When the source passes through a point $\bu_0$ on a fold arc
abutting the cusp, the image centroid has a jump discontinuity; we present a
formula for the size of the jump in terms of the local derivatives of the lens
potential. The total magnifications for a small extended source located both
on, and perpendicular to, the axis of symmetry are also derived, for both
uniform and limb darkened surface brightness profiles. We find that the
difference in magnification between a finite and point source is $< 5%$ for
separations of $> 2.5$ source radii from the cusp point, while the effect of
limb-darkening is $< 1%$ in the same range. Our predictions for the astrometric
and photometric behavior of both pointlike and finite sources passing near a
cusp are illustrated and verified using numerical simulations of the
cusp-crossing Galactic binary-lens event MACHO-1997-BUL-28. We discuss several
possible applications of our results to Galactic binary lenses and quasar
microlensing.
