  The first static spherically symmetric perfect fluid solution with constant
density was found by Schwarzschild in 1918. Generically, perfect fluid spheres
are interesting because they are first approximations to any attempt at
building a realistic model for a general relativistic star. Over the past 90
years a confusing tangle of specific perfect fluid spheres has been discovered,
with most of these examples seemingly independent from each other. To bring
some order to this collection, 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. In addition, we develop new ``solution
generating'' theorems for the TOV, whereby any given solution can be
``deformed'' to a new solution. Because these TOV-based theorems work directly
in terms of the pressure profile and density profile it is relatively easy to
impose regularity conditions at the centre of the fluid sphere.
