The two main results of the article are concerned with Anderson Localization
for one-dimensional lattice Schroedinger operators with quasi-periodic
potentials with d frequencies. First, in the case d = 1 or 2, it is proved that
the spectrum is pure-point with exponentially decaying eigenfunctions for all
potentials (defined in terms of a trigonometric polynomial on the d-dimensional
torus) for which the Lyapounov exponents are strictly positive for all
frequencies and all energies. Second, for every non-constant real-analytic
potential and with a Diophantine set of d frequencies, a lower bound is given
for the Lyapounov exponents for the same potential rescaled by a sufficiently
large constant.