While attempts have been made to solve exactly Eqs. ( ) and ( ) , the most interesting developments of the exact factorization of the electron-nuclear wavefunction focused on introducing the quantum-classical perspective within this framework, such that the nuclear dynamics is ultimately approximated using classical-like trajectories while the electronic dynamics is treated quantum mechanically . The quantum-classical perspective naturally emerges in the exact factorization, since the classical limit of Eq. ( ) simply means to interpret the TDVP and TDPES as standard (classical) electromagnetic potentials producing a classical force on the nuclei 109 . The main efforts in this context have been devoted, then, to the calculation and approximation of the TDVP and of the TD-PES from the solution of the quantum-mechanical electronic equation (5). To this end, extensive work has been devoted to analyze the TDVP and TDPES in model situations, such as in photo-activated processes , in relaxation dynamics through conical intersections , in the presence of quantum interferences 178 , under the effect of an external time-dependent field , or including spin-orbit coupling . Those numerical analyses, together with theoretical developments focusing on understanding the behavior of the equations in some limiting cases, e.g, the adiabatic limit 184 , or on resolution strategies of the nuclear equation, e.g., using semiclassical techniques 185 or the method of characteristics , yielded various trajectorybased schemes readily applicable to on-the-fly moleculardynamics simulations, as we will present below.