  We discuss the theory of mixtures of Bosonic and Fermionic atoms in periodic
potentials at zero temperature. We derive a general Bose--Fermi Hubbard
Hamiltonian in a one--dimensional optical lattice with a superimposed harmonic
trapping potential. We study the conditions for linear stability of the mixture
and derive a mean field criterion for the onset of a Bosonic superfluid
transition. We investigate the ground state properties of the mixture in the
Gutzwiller formulation of mean field theory, and present numerical studies of
finite systems. The Bosonic and Fermionic density distributions and the onset
of quantum phase transitions to demixing and to a Bosonic Mott--insulator are
studied as a function of the lattice potential strength. The existence is
predicted of a disordered phase for mixtures loaded in very deep lattices. Such
a disordered phase possessing many degenerate or quasi--degenerate ground
states is related to a breaking of the mirror symmetry in the lattice.
