The abstract theory of equilibrium problems has found its applications in various branches of nonlinear sciences. It provides us a unified structure to study an extensive number of linear and nonlinear physical problems. Since the projection method cannot be used to solve equilibrium problems, so we use auxiliary principle technique to solve equilibrium problems. Our main focus is to apply this technique to suggest several iterative algorithms (implicit and explicit) for solving different classes of equilibrium problems and their variant forms. Some other algorithms are suggested using the combination of auxiliary principle technique and so-called Bregman function. We also study the convergence analysis of these methods. We also consider and analyze a class of regularized mixed quasi variational inequalities. The auxiliary principle technique is again used to suggest some new iterative methods for solving the mixed quasi variational inequalities involving the skew symmetric bifunction. The convergence criterion of these iterative methods for solving the mixed quasi variational inequalities is considered under suitable conditions. Several special cases of these problems are considered. Results proved in this thesis continue to hold for known and new classes of equilibrium problems.