We review the dynamical properties of a Bose-Einstein condensate (BEC) trapped in one-dimensional periodic potentials. We study the Bloch and the Bogoliubov excitation spectrums introducing two different effective masses, and we investigate the Sound-wave and the Newtonian regimes of the system. We finally discuss the breakdown of superfluidity due to the onset of dynamical instabilities, focusing on the dephasing of Josephson oscillations, pinning, and thermalization.