In this paper, we investigate the logical strength of two types of fixed point theorems in the context of reverse mathematics. One is concerned with extensions of the Banach contraction principle. Among theorems in this type, we mainly show that the Caristi fixed point theorem is equivalent to ACA over RCA0 . The other is dedicated to topological fixed point theorems such as the Brouwer fixed point theorem. We introduce some variants of the Fan-Browder fixed point theorem and the Kakutani fixed point theorem, which we call FBFP and KFP, respectively. Then we show that FBFP is equivalent to WKL and KFP is equivalent to ACA, over RCA0 . In addition, we also study the application of the Fan-Browder fixed point theorem to game systems.