  We provide a complete and self-contained proof of spectral and dynamical
localization for the one-dimensional Anderson model, starting from the
positivity of the Lyapunov exponent provided by F\"urstenberg's theorem. That
is, a Schr\"odinger operator in $\ell^2(\mathbb{Z})$ whose potential is given
by independent identically distributed (i.i.d.) random variables almost surely
has pure point spectrum with exponentially decaying eigenfunctions and its
unitary group exhibits exponential off-diagonal decay, uniformly in time. This
is achieved by way of a new result: for the Anderson model, one typically has
Lyapunov behavior for all generalized eigenfunctions. We also explain how to
obtain analogous statements for extended CMV matrices whose Verblunsky
coefficients are i.i.d., as well as for half-line analogs of these models.
