Searching for Supersymmetry: The $\mu\nu$SSM

We review the role played by the '$\mu$ from $\nu$' supersymmetric standard model ($\mu \nu$SSM) in the search for supersymmetry. First, we discuss its theoretical motivation, that is the simultaneous solution of $\mu$- and $\nu$-problems through the introduction of right-handed neutrinos. The latter produces $R$-parity violation (RPV), giving rise to interesting signals of new physics. As by-products, in the $\mu\nu$SSM there are dark matter candidates, and electroweak baryogenesis can be realized. Then, we survey signals by which the model could be tested at the large hadron collider (LHC). In addition to the enlarged Higgs sector with sneutrinos, we put special emphasis in analyzing the intimate connection between the lightest supersymmetric particle (LSP) lifetime and the size of neutrino Yukawa couplings. Displaced vertices and/or multileptons are some of the interesting signatures that can be probed. Finally, we discuss possible extensions of the $\mu\nu$SSM such as the inclusion in the superpotential of the conventional trilinear lepton-number violating couplings, the addition of an extra $U(1)'$ gauge group to the symmetry of the standard model, or the reinterpretation of the Higgs doublets as a fourth family of leptons superfields motivating the existence of a fourth family of vector-like quark doublet superfields.