  We consider a novel class of $f(\R)$ gravity theories where the connection is
related to the conformally scaled metric $\hat g_{\mu\nu}=C(\R)g_{\mu\nu}$ with
a scaling that depends on the scalar curvature $\R$ only. We call them
C-theories and show that the Einstein and Palatini gravities can be obtained as
special limits. In addition, C-theories include completely new physically
distinct gravity theories even when $f(\R)=\R$. With nonlinear $f(\R)$,
C-theories interpolate and extrapolate the Einstein and Palatini cases and may
avoid some of their conceptual and observational problems. We further show that
C-theories have a scalar-tensor formulation, which in some special cases
reduces to simple Brans-Dicke-type gravity. If matter fields couple to the
connection, the conservation laws in C-theories are modified. The stability of
perturbations about flat space is determined by a simple condition on the
lagrangian.
