IN the last fifteen years, variational inequalities have gainedimportance in analysis, both from the theoretical and the practical pointsof view. At the beginning of this work we emphasize the formulation ofvarious types of problems in a variational setting. We deal with variousmathematical aspects of many of these special variational formulationsand with the numerical analysis for some of these problems. To completethis work; we deal with a variety of aspects of these subjects: thedevelopment of different types of contact problems; the detailedformulation of appropriate variational principles; the identification of thespaces in which these formulations make sense; when possible, theestablishment of sufficient conditions for the existence and uniqueness ofsolutions; the proof of convergence theorems and a priori error estimatesfor finite element approximations; the description of numerical schemesfor solving discrete variational inequalities, including the actualnumerical solutions of many representative example problems, and finallythe interpretation of these results with regard to their implications on the. actual physics of the problems considered.This thesis consists of six chapters:Chapter one: addresses historical review on the variationalinequalities and then record some notation, theories that are usedthroughout this thesis.