  General Relativity gives that finitely many point masses between an observer
and a light source create many images of the light source. Positions of these
images are solutions of $r(z)=\bar{z},$ where $r(z)$ is a rational function. We
study the number of solutions to $p(z) = \bar{z}$ and $r(z) = \bar{z},$ where
$p(z)$ and $r(z)$ are polynomials and rational functions, respectively. Upper
and lower bounds were previously obtained by Khavinson-\'{S}wi\c{a}tek,
Khavinson-Neumann, and Petters. Between these bounds, we show that any number
of simple zeros allowed by the Argument Principle occurs and nothing else
occurs, off of a proper real algebraic set. If $r(z) = \bar{z}$ describes an
$n$-point gravitational lens, we determine the possible numbers of generic
images.
