  For a fairly general family of L-functions, we survey the known consequences
of the existence of asymptotic formulas with power-sawing error term for the
(twisted) first and second moments of the central values in the family. We then
consider in detail the important special case of the family of twists of a
fixed cusp form by primitive Dirichlet characters modulo a prime q, and prove
that it satisfies such formulas. We derive arithmetic consequences:
  - a positive proportion of central values L(f x chi, 1/2) are non-zero, and
indeed bounded from below;
  - there exist many characters chi for which the central L-value is very
large;
  - the probability of a large analytic rank decays exponentially fast.
  We finally show how the second moment estimate establishes a special case of
a conjecture of Mazur and Rubin concerning the distribution of modular symbols.
