  We survey what is known about minimal surfaces in $\bold R^3 $ that are
complete, embedded, and have finite total curvature. The only classically known
examples of such surfaces were the plane and the catenoid. The discovery by
Costa, early in the last decade, of a new example that proved to be embedded
sparked a great deal of research in this area. Many new examples have been
found, even families of them, as will be described below. The central question
has been transformed from whether or not there are any examples except surfaces
of rotation to one of understanding the structure of the space of examples.
