  These notes outline recent developments in classical minimal surface theory
that are essential in classifying the properly embedded minimal planar domains
M in R^3 with infinite topology (equivalently, with an infinite number of
ends). This final classification result by Meeks, Perez, and Ros states that
such an M must be congruent to a homothetic scaling of one of the classical
examples found by Riemann in 1860. These examples {\cal R}_s, 0<s<\infty, are
defined in terms of the Weierstrass {\cal P}-functions {\cal P}_t on the
rectangular elliptic curve {\C} / {< 1, t\sqrt{-1}>}, are singly-periodic and
intersect each horizontal plane in R^3 in a circle or a line parallel to the
x-axis. Earlier work by Collin, Lopez and Ros and Meeks and Rosenberg
demonstrate that the plane, the catenoid and the helicoid are the only properly
embedded minimal surfaces of genus zero with finite topology (equivalently,
with a finite number of ends). Since the surfaces {\cal R}_s converge to a
catenoid as s tends to 0 and to a helicoid as s tends to infinity, then the
moduli space {\cal M} of all properly embedded, non-planar, minimal planar
domains in R^3 is homeomorphic to the closed unit interval [0,1].
