  Near the end of his life, Bernhard Riemann made the marvelous discovery of a
1-parameter family $R_{\lambda}$, $\lambda\in (0,\infty)$, of periodic properly
embedded minimal surfaces in $\mathbb{R}^3$ with the property that every
horizontal plane intersects each of his examples in either a circle or a
straight line. Furthermore, as the parameter $\lambda\to 0$ his surfaces
converge to a vertical catenoid and as $\lambda\to \infty$ his surfaces
converge to a vertical helicoid. Since Riemann's minimal examples are
topologically planar domains that are periodic with the fundamental domains for
the associated $\mathbb{Z}$-action being diffeomorphic to a compact annulus
punctured in a single point, then topologically each of these surfaces is
diffeomorphic to the unique genus zero surface with two limit ends. Also he
described his surfaces analytically in terms of elliptic functions on
rectangular elliptic curves. This article exams Riemann's original proof of the
classification of minimal surfaces foliated by circles and lines in parallel
planes and presents a complete outline of the recent proof that every properly
embedded minimal planar domain in $\mathbb{R}^3$ is either a Riemann minimal
example, a catenoid, a helicoid or a plane.
