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Electron-molecule collisions are described by Hamiltonians with continuum energy spectra, in contrast with conventional electronic structure problems which have discrete energy levels. Since computers can only handle discrete matrix algebra, the simulation of electron scattering by molecules is particularly challenging. We will present an overview of the available computational approaches to electronic collisions as well as their applications to contemporary problems in materials and life sciences. Special attention will be paid to low energy collisions and electron attachment. The formation of resonances (temporary negative ions) is known to initiate vibration dynamics that often leads to the dissociation of the molecular anions. This process, known as dissociative electron attachment (DEA), poses additional challenges to theory, since the continuum electronic problem is now coupled to the vibrational continuum defined by the dissociation problem. We will also discuss the recent adaptation of numerical methods developed for photo-induced nuclear dynamics to DEA dynamics.