The book is devoted to a rapidly developing part of modern theoretical physics - the theory of quantum transport in nanosystems. The theoretical methods considered in the book are in the basis of our understanding of electron transport at nanoscale, and are widely used in nanoelectronics, molecular electronics and spin-dependent electronics (spintronics). The book is based on lectures for graduate and post-graduate students in the University of Regensburg. The first part is devoted to the basic concepts of nanotransport: Landauer-Buttiker method and matrix Green function method for coherent transport, master equation method for tunneling, Coulomb blockade, vibrons and polarons. The results in this part are obtained without sophisticated techniques, such as the generalized quantum master equation and nonequilibrium Green functions, which are considered in the second part. The third part is devoted to an introduction into the single-molecule transport theory.