Newton’s “tangent method” for solving algebraic equation and Picard’s iterative method for solving initial value problems of ordinary differential equation are the original forms of Banach contraction mapping principle. In the beginning of twentieth century, due to the establishment of the concepts of infinite-dimensional spaces and operators, the existence problem for the solutions of many algebraic equations, differential equations, integral equations, and functional equations can be unified and transformed into the existence of fixed points of operators in suitable infinite-dimensional spaces. The existence of saddle points for some minimax problems can be transformed into the existence of fixed points of the continuous mappings from closed convex sets in finite-dimensional space to themselves. The existence problem of periodic solutions of restricted three-body problems can be transformed into the existence problem of fixed points of the homeomorphic mapping of the planar circular annulus to itself with area-preserving and boundary twisting. Therefore, the existence and the number and calculation methods of fixed points of various forms of mappings sometimes appear very important. However, due to space limitations, this chapter only selects five of the most classic and important fixed point theorems for a brief introduction, for a more comprehensive and in-depth study of these fixed point theorems and their applications, see Granas A. and Dugundji J.’s Monograph “Fixed Point Theory”, Springer, 2003.