  This is my PhD thesis submitted to the Weizmann Institute of Science. It is
based on the papers [AG08c], [AG08d], [AGRS07], [AGS08], [AGS09], [Aiz08] and
[SZ08].
  This thesis includes an introduction to Gelfand pairs and invariant
distributions, a list of tools to work with invariant distributions oriented
towards proving Gelfand property and a proof that the pair (GL(n+1,F),GL(n,F))
is a strong Gelfand pair.
  Namely, we prove that if $\pi$ is an irreducible admissible smooth
representation of GL(n+1,F) and $\rho$ is an irreducible admissible smooth
representation of GL(n,F) then
  $$dim Hom_{GL(n,F)}(\pi,\rho)\leq 1.$$
