  We present an overview of beyond mean field theories (BMFT) based on the
generator coordinate method (GCM) and the recovery of symmetries used in
nuclear physics with effective forces. After a reminder of the
Hartree-Fock-Bogoliubov (HFB) theory a discussion of the shortcomings of any
mean field approximation (MFA) is presented. The recovery of the symmetries
spontaneously broken in the HFB approach, in particular the angular momentum,
is necessary, among others, to describe excited states and transitions.
Particle number projection is needed to guarantee the right number of protons
and neutrons. Furthermore a projection before the variation prevents the
pairing collapse in the weak pairing regime. The lack of fluctuations around
the average values of the MFA is a shortcoming of this approach. To build in
correlations in BMFT one selects the relevant degrees of freedom: quadrupole,
octupole and the pairing vibrations as well as the single particle ones. In the
GCM the operators representing these degrees of freedom are used as coordinates
to generate a collective subspace. The highly correlated GCM wave function is
finally written as a linear combination of a projected basis of this space. The
variation of the coefficients of the linear combination leads to the
Hill-Wheeler equation. We discuss the classical beta and gamma vibrations by
considering the quadrupole operators as coordinates. We present pairing
fluctuations by considering the pairing gaps as generator coordinates. Lastly
the explicit consideration of the time reversal symmetry breaking in the HFB
wave function by the cranking procedure allows the alignment of nucleon pairs
opening a new dimension in the BMFT calculations. Abundant calculations with
the Gogny force illustrate the state-of-the-art of BMFTs with density
functionals. We conclude with a thorough discussion on the potential poles of
the theory.
