In this manuscript, we provide an introduction to those topics of fixed point theory that is one of the most important and powerful tools of the modern mathematics not only it is used on a daily basis in pure and applied mathematics but it is also solving a bridge between analysis and topology and provide a very fruitful are of interaction between the two. The theory of fixed points belongs to topology, a part of mathematics created at the end of the nineteenth century. The famous French Mathematician H. Poincare (1854-1912) was the founder of the fixed point approach. He had deep insight into its future importance for problems of mathematical analysis and celestial mechanics and took an active part in its development. In spite of their elementary character, the results given here have a number of significant applications. Some of these are presented at the end of the chapter.