Mathematical models in contact mechanics Contact processes between deformable bodies abound in industry and everyday life and for this reason considerable efforts have been made in their modelling and analysis. Owing to their inherent complexity contact phenomena lead to new and interesting mathematical models. Here and everywhere in this book by a mathematical model we mean a system of partial differential equations associated with boundary conditions and initial conditions eventually which describes a specific contact process. The purpose of this book is to introduce the reader to some representative mathematical models which arise in Contact Mechanics. Our aim is twofold: first to present a sound and rigorous description of the way in which the mathematical models are constructed; second to present the mathematical analysis of such models which includes the variational formulation existence uniqueness and convergence results. To this end we use results on various classes of variational inequalities in Hilbert spaces that we present in an abstract functional framework. Also we use various functional methods including monotonicity compactness penalization regularization and duality methods. Moreover we pay particular attention to the mechanical interpretation of our results and in this way we illustrate the cross fertilization between modelling and applications on the one hand and nonlinear analysis on the other hand This text provides a complete introduction to the theory of variational inequalities with emphasis on contact mechanics. It covers existence uniqueness and convergence results for variational inequalities including the modelling and variational analysis of specific frictional contact problems with elastic viscoelastic and viscoplastic materials. New models of contact are presented including contact of piezoelectric materials. Particular attention is paid to the study of history dependent quasivariational inequalities and to their applications in the study of contact problems with unilateral constraints. The book fully illustrates the cross fertilisation between modelling and applications on the one hand and nonlinear mathematical analysis on the other. Indeed the reader will gain an understanding of how new and nonstandard models in contact mechanics lead to new types of variational inequalities and conversely how abstract results concerning variational inequalities can be applied to prove the unique solvability of the corresponding contact problems