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Tribology is the science of friction, lubrication and wear of moving components. Results obtained from tribology are used to reduce energy losses in friction pro- cesses, to reduce material losses due to wear, and to increase the service life of components. Contact Mechanics plays an important role in Tribology. Contact Mechanics studies the stress and strain states of bodies in contact; it is contact that leads to friction interaction and wear. This book investigates a variety of contact problems: discrete contact of rough surfaces, the effect of imperfect elasticity and mechanical inhomogeneity of contacting bodies, models of friction and wear, changes in contact characteristics during the wear process, etc. The results presented in this book were obtained during my work at the Insti- tute for Problems in Mechanics of the Russian Academy of Sciences. The first steps of this research were carried out under the supervision of Professor L. A. Galin who taught me and showed me the beauty of scientific research and solutions. Some of the problems included in the book were investigated together with my col- leagues Dr. M. N. Dobychin, Dr. O. G. Chekina, Dr. I. A. Soldatenkov, and Dr. E. V. Tor- skaya from the Laboratory of Friction and Wear (IPM RAS) and Prof. F. Sadeghi from Purdue University (West Lafayette, USA). I would like to express my thanks to them. I am very grateful to Professor G. M. L.