  We review our studies on Bose and Fermi superfluids of cold atomic gases in
optical lattices at zero temperature. Especially, we focus on superfluid Fermi
gases along the crossover between the Bardeen-Cooper-Schrieffer (BCS) and the
Bose-Einstein condensate (BEC) states, which enable us to study the Bose and
the Fermi superfluids in a unified point of view. We discuss basic static and
long-wavelength properties (such as the equation of state, incompressibility,
and effective mass), energetic stability, and energy band structures of the
superfluid Fermi gases in an optical lattice periodic along one spatial
direction. The periodic potential causes pairs of atoms to be strongly bound,
and this can affect the static and long-wavelength properties and the stability
of the superflow. Regarding the band structure, a peculiar loop structure
called "swallowtail" can appear in superfluid Fermi gases and in the Bose case,
but the mechanism of emergence in the Fermi case is very different from that in
bosonic case. Other quantum phases that the cold atomic gases in optical
lattices can show are also briefly discussed based on their roles as quantum
simulators of Hubbard models.
