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We present an overview of beyond mean field theories (BMFT) based on thegenerator coordinate method (GCM) and the recovery of symmetries used innuclear physics with effective forces. After a reminder of theHartree-Fock-Bogoliubov (HFB) theory a discussion of the shortcomings of anymean field approximation (MFA) is presented. The recovery of the symmetriesspontaneously 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 protonsand neutrons. Furthermore a projection before the variation prevents thepairing collapse in the weak pairing regime. The lack of fluctuations aroundthe average values of the MFA is a shortcoming of this approach. To build incorrelations in BMFT one selects the relevant degrees of freedom: quadrupole,octupole and the pairing vibrations as well as the single particle ones. In theGCM the operators representing these degrees of freedom are used as coordinatesto generate a collective subspace. The highly correlated GCM wave function isfinally written as a linear combination of a projected basis of this space. Thevariation of the coefficients of the linear combination leads to theHill-Wheeler equation. We discuss the classical beta and gamma vibrations byconsidering the quadrupole operators as coordinates. We present pairingfluctuations by considering the pairing gaps as generator coordinates. Lastlythe explicit consideration of the time reversal symmetry breaking in the HFBwave function by the cranking procedure allows the alignment of nucleon pairsopening a new dimension in the BMFT calculations. Abundant calculations withthe Gogny force illustrate the state-of-the-art of BMFTs with densityfunctionals. We conclude with a thorough discussion on the potential poles ofthe theory.

State-of-the-art of beyond mean field theories with nuclear density functionals