  We present a survey of ergodic theorems for actions of algebraic and
arithmetic groups recently established by the authors, as well as some of their
applications. Our approach is based on spectral methods employing the unitary
representation theory of the groups involved. This allows the derivation of
ergodic theorems with a rate of convergence, an important phenomenon which does
not arise in classical ergodic theory. Quantitative ergodic theorems give rise
to new and previously inaccessible applications, and we demonstrate the
remarkable diversity of such applications by presenting several
number-theoretic results. These include, in particular, general uniform error
estimates in lattice points counting problems, explicit estimates in sifting
problems for almost-prime points on symmetric varieties, bounds for exponents
of intrinsic Diophantine approximation, and results on fast distribution of
dense orbits on homogeneous spaces.
