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The work studies, by means of ab-initio (from-the-first-principles) calculations, effects of the spin-orbit interaction on electron-transport properties of both purely metallic and tunnelling magnetic multilayers. It is shown that even in systems composed of relatively light elements such as nickel and cobalt the interaction may play a crucial role, and not including it in the calculation framework may result in erroneous predictions. The book is written in a self-contained and comprehensive manner. It starts with a detailed explanation of the density-functional theory (DFT), then the particular method of linear muffin-tin orbitals (LMTO) is reviewed. Regarding transport, a form of a Landauer-like formula is derived. Finally, the theory is applied on illustrative systems which show a particularly marked effects of the spin-orbit interaction.