  In 1997, Collin proved that any properly embedded minimal surface in
$\mathbb{R}^3$ with finite topology and more than one end has finite total
Gaussian curvature. Hence, by an earlier result of Lopez and Ros, catenoids are
the only non-planar, non-simply connected, properly embedded, minimal planar
domains in $\mathbb{R}^3$ of finite topology. In 2005, Meeks and Rosenberg
proved that the only simply connected, properly embedded minimal surfaces in
$\mathbb{R}^3$ are planes and helicoids. Around 1860, Riemann defined a
one-parameter family of periodic, infinite topology, properly embedded, minimal
planar domains $\mathcal{R}_t$ in $\mathbb{R}^3$, $t\in (0,\infty )$. These
surfaces are called the Riemann minimal examples, and the family $\{
\mathcal{R}_t\} _t$ has natural limits being a vertical catenoid as $t\to 0$,
and a vertical helicoid as $t\to \infty $. In this paper we complete the
classification of properly embedded, minimal planar domains in $\mathbb{R}^3$
by proving that the only connected examples with infinite topology are the
Riemann minimal examples. We also prove that the limit ends of Riemann minimal
examples are model surfaces for the limit ends of properly embedded minimal
surfaces $M\subset \mathbb{R}^3$ of finite genus and infinite topology, in the
sense that such an $M$ has two limit ends, each of which has a representative
which is naturally asymptotic to a limit end representative of a Riemann
minimal example with the same associated flux vector.
