  We develop an accurate theory of resonantly interacting Fermi mixtures with
both spin and mass imbalance. We consider Fermi mixtures with arbitrary mass
imbalances, but focus in particular on the experimentally available
${}^{6}$Li-${}^{40}$K mixture. We determine the phase diagram of the mixture
for different interactions strengths that lie on the BCS side of the Feshbach
resonance. We also determine the universal phase diagram at unitarity. We find
for the mixtures with a sufficiently large mass imbalance, that includes the
${}^{6}$Li-${}^{40}$K mixture, a Lifshitz point in the universal phase diagram
that signals an instability towards a supersolid phase.
