The term ‘Metric’ Fixed Point Theory refers to those fixed point theoretic results in which geometric conditions on the underlying spaces and/or mappings play a crucial role. Obviously there can be no clear line separating this branch of fixed point theory from either the topological or set-theoretic branches since metric methods are often useful in proving results which are basically nonmetric in nature, and vice versa. However, the results considered here are always couched in at least a metric space framework, usually in a Banach space setting, and the methods typically involve both the topological and the geometric structure of the space in conjunction with metric constraints on the behavior of the mappings.

For the past twenty-five years metric fixed point theory has been a flourishing area of research for many mathematicians. Although a substantial number of definitive results have now been discovered, a few questions lying at the heart of the theory remain open and there are many unanswered questions regarding the limits to which the theory may be extended. Some of these questions are merely tantalizing while others suggest substantial new avenues of research.

It is apparent that the theory has now reached a level of maturity appropriate to an examination of its central themes. The topics selected for this text were chosen accordingly. No attempt has been made to explore all aspects of the theory nor to present a compendium of known facts.