  We investigate many-body phase diagrams of atomic boson-fermion mixtures
loaded in the two-dimensional optical lattice. Bosons mediate an attractive,
finite-range interaction between fermions, leading to fermion pairing phases of
different orbital symmetries. Specifically, we show that by properly tuning
atomic and lattice parameters it is possible to create superfluids with $s$-,
$p$-, and d-wave pairing symmetry as well as spin and charge density wave
phases. These phases and their stability are analyzed within the mean-field
approximation for systems of $^{40}$K-$^{87}$Rb and $^{40}$K-$^{23}$Na
mixtures. For the experimentally accessible regime of parameters, superfluids
with unconventional fermion pairing have transition temperature around a
percent of the Fermi energy.
