  Motivated by recent surprising experimental findings, we develop a
strong-coupling theory for Bose-Fermi mixtures capable of treating resonant
inter-species interactions while satisfying the compressibility sum rule. We
show that the mixture can be stable at large interaction strengths close to
resonance, in agreement with the experiment but at odds with the widely used
perturbation theory. We also calculate the sound velocity of the Bose gas in
the $^{133}$Cs-$^6$Li mixture, again finding good agreement with the
experimental observations both at weak and strong interactions. A central
ingredient of our theory is the generalization of a fermion mediated
interaction to strong Bose-Fermi scatterings and to finite frequencies. This
further leads to a predicted hybridization of the sound modes of the Bose and
Fermi gases, which can be directly observed using Bragg spectroscopy.
