ixed point theory is most often associated with the task of solving equations. Indeed, many problems in biology, economics, physics, and engineering involve equations and mathematics is the one to provide methods for solving them. But any equation of the form f(x) = 0 can be transformed into a new problem by a simple transformation: G(x) = f(x)+x = x. Points that remains invariant under the action of a function are called fixed points. When interested in the existence of solutions to generalized real life problems, the question can be translated into asking about whether the problem has fixed points. The work reported in this thesis is based on fixed points and common fixed points of a variety of maps satisfying different contractive conditions. The fixed point in generalized metric spaces is investigated in a new setting. Several existing results concerning single-valued and multi-valued maps are extended in symmetric spaces. Some fixed point theorems for Lipschitzian mappings on a metric space with a uniform normal structure are also established...