Atypical trajectories often play a crucial roles in dynamical systems. For instance, resonances and separatrices determine the fate of planetary systems, and localized objects like solitons and breathers provide mechanisms of energy transport in nonlinear systems such as Bose-Einstein condensates and biological molecules. I will present a simulation method to simulate such rare trajectories in an efficient, accelerated way and show how this method gives access to central quantities in non-equilibrium statistical mechanics: the large deviation functions. \ par The first part of my talk will focus on applications in the field of dynamical systems whereas the second part will deal with standard problems of non-equilibrium statistical mechanics, such as exclusion processes and absorbing phase transitions.