  We investigate a cold atomic mixture of spinless bosons and fermions in
two-dimensional optical lattices. In the presence of a nested Fermi surface,
the bosons may develop a fascinating supersolid behavior characterized by a
finite superfluid density as well as a spatial density wave order. Focusing on
the triangular lattice geometry and combining a general Landau-Ginzburg-Wilson
approach with microscopically derived mean-field theory, we find an exotic
supersolid phase at a fermionic band-filling of $n_f = 3/4$ with a Kagome-type
crystalline order. We also address the case of anisotropic hopping amplitudes,
and show that striped supersolid phases emerge on the square and triangular
lattices. For weak interactions, the supersolid competes with phase separation.
For strong intra- and inter-species interactions, with the total number of
fermions and bosons corresponding to one particle per site, the bosons form an
alternating Mott insulator ground state. Finally, for a mixture of
$^{87}\text{Rb}^{40}\text{K}$ and $^{23}\text{Na}^6\text{Li}$, we show that
supersolidity can be observed in the range of accessible temperatures in the
square lattice geometry.
