8 pags., 2 figs. We introduce electron energy-gain spectroscopy as a tool to yield information on local optical excitations of nanostructured systems using transmission electron microscopes equipped with external optical illumination. The new spectroscopy combines the superb spatial resolution of electron microscopes with unprecedented energy resolution below the milli-electron-volt level, only limited by the bandwidth of the external light. The analysis of energy gain events should reveal hyperfine details in the optical response of individual nanostructures (e.g. plasmons in nanoparticles). Our conclusions rely on a general formalism capable of describing light absorption by fast electrons moving in vacuum near an illuminated nanostructure, thus paving the way towards new light-assisted electron- and ion-acceleration schemes. Energy gain probabilities are shown to be comparable to those observed in energy loss experiments for reasonable illumination intensity. This work was supported in part by the Spanish MEC (MAT2007-66050) and by the EU (STRP-016881-SPANS). Peer reviewed