  It has been shown that univalent circle packings filling in the complex plane
$\bold C$ are unique up to similarities of $\bold C$. Here we prove that
bounded degree branched circle packings properly covering $\bold C$ are
uniquely determined, up to similarities of $\bold C$, by their branch sets. In
particular, when branch sets of the packings considered are empty we obtain the
earlier result.
  We also establish a circle packing analogue of Liouville's theorem: if $f$ is
a circle packing map whose domain packing is infinite, univalent, and has
recurrent tangency graph, then the ratio map associated with $f$ is either
unbounded or constant.

Metric Geometry;Complex Variables