  We study the renormalization of theories of gravity with an arbitrary
(torsionful and non-metric) connection. The class of actions we consider is of
the Palatini type, including the most general terms with up to two derivatives
of the metric, but no derivatives of the connection. It contains 19 independent
parameters. We calculate the one loop beta functions of these parameters and
find their fixed points. The Holst subspace is discussed in some detail and
found not to be stable under renormalization. Some possible implications for
ultraviolet and infrared gravity are discussed.
