  Ever since Karl Schwarzschild's 1916 discovery of the spacetime geometry
describing the interior of a particular idealized general relativistic star --
a static spherically symmetric blob of fluid with position-independent density
-- the general relativity community has continued to devote considerable time
and energy to understanding the general-relativistic static perfect fluid
sphere. Over the last 90 years a tangle of specific perfect fluid spheres has
been discovered, with most of these specific examples seemingly independent
from each other. To bring some order to this collection, in this article we
develop several new transformation theorems that map perfect fluid spheres into
perfect fluid spheres. These transformation theorems sometimes lead to
unexpected connections between previously known perfect fluid spheres,
sometimes lead to new previously unknown perfect fluid spheres, and in general
can be used to develop a systematic way of classifying the set of all perfect
fluid spheres.
