  Minimal surfaces with uniform curvature (or area) bounds have been well
understood and the regularity theory is complete, yet essentially nothing was
known without such bounds. We discuss here the theory of embedded (i.e.,
without self-intersections) minimal surfaces in Euclidean 3-space without a
priori bounds. The study is divided into three cases, depending on the topology
of the surface. Case one is where the surface is a disk, in case two the
surface is a planar domain (genus zero), and the third case is that of finite
(non-zero) genus. The complete understanding of the disk case is applied in
both cases two and three.
  As we will see, the helicoid, which is a double spiral staircase, is the most
important example of an embedded minimal disk. In fact, we will see that every
such disk is either a graph of a function or part of a double spiral staircase.
The helicoid was discovered to be a minimal surface by Meusnier in 1776.
  For planar domains the fundamental examples are the catenoid, also discovered
by Meusnier in 1776, and the Riemann examples discovered by Riemann in the
beginning of the 1860s. Finally, for general fixed genus an important example
is the recent example by Hoffman-Weber-Wolf of a genus one helicoid.
  In the last section we discuss why embedded minimal surfaces are
automatically proper. This was known as the Calabi-Yau conjectures for embedded
surfaces. For immersed surfaces there are counter-examples by Jorge-Xavier and
Nadirashvili.
