Hardy Gross Max Planck Institute of Microstructure Physics The coupling between electronic and nuclear motion beyond the Born-Oppenheimer approximation plays an important role in a variety of phenomena. Prominent examples are the process of vision, photosynthesis, as well as superconductivity. In this lecture, a novel concept of time-dependent potential energy surfaces [1,2] will be presented. We deduce an exact factorization of the complete wavefunction into a purely nuclear part and a many-electron wavefunction which parametrically depends on the nuclear configuration. We derive equations of motion for the nuclear and electronic wavefunctions. The nuclear factor satisfies a time-dependent Schroedinger equation featuring a vector potential and an (N-body) potential energy surface. These potentials are unique (up to within a gauge transformation) and are the "correct" potentials in the sense that they give a purely nuclear wavefunction yielding the true nuclear N-body density and the true nuclear N-body current density of the full electron-nuclear system. Hence, in the classical limit, the gradient of this exact potential energy surface yields the "correct" classical force on the nuclei. With some simple examples we demonstrate the significance of these concepts in understanding the full electron-ion dynamics. In particular, whenever there is a splitting of the exact nuclear wavepacket in the vicinity of an avoided crossing, the exact time-dependent surface shows a nearly discontinuous step [3], making the classical force on the nuclei jump from one to another adiabatic surface. This observation, which is reminiscent of Tully's algorithm of surface hopping, suggests a novel mixed-quantum-classical algorithm to treat the coupled electron-nuclear motion. [1] A. Abedi, N.T. Maitra, E.K.U. Gross, PRL 105, 123002 (2010). [2] A. Abedi, N.T. Maitra, E.K.U. Gross, JCP 137, 22A530 (2012). [3] A. Abedi, F. Agostini, Y. Suzuki, E.K.U. Gross, PRL 110, 263001 (2013)