This thesis describes theoretical and numerical investigations of inelastic scat- tering and energy dissipation in electron transport through nanoscale sys- tems. A computational scheme, based on a combination of density functional theory (DFT) and nonequilibrium Green’s functions (NEGF), has been devel- oped to describe the electrical conduction properties taking into account the full atomistic details of the systems. The scheme involves quantitative calcu- lations of electronic structure, vibrational modes and frequencies, electron- vibration couplings, and inelastic current-voltage characteristics in the weak coupling limit. When a current is passed through a nanoscale device, such as a single molecule or an atomic-size contact, it will heat up due to excitations of the nuclear vibrations. The developed scheme is able to quantify this local heating effect and to predict how it affects the conductance. The methods have been applied to a number of specific systems, includ- ing monatomic gold chains, atomic point contacts, and metal-molecule-metal configurations. These studies have clarified the inelastic effects in the elec- tron transport and characterized the vibrational modes that couple to the current. For instance, the dominant scattering for gold chains could be traced back to the longitudinal “alternating bond-length” mode. Furthermore, the results have been compared critically with experimental measurements for the different systems, and provided a microscopic understanding for the im- portant physics. An example is the current-induced fluctuations that have been shown to influence the transport though individual C60 molecules on copper surfaces.