We extend the classical primal-dual interior point algorithms from the Euclidean setting to the Riemannian one. Our method, named the Riemannian interior point (RIP) method, is for solving Riemannian constrained optimization problems. Under the standard assumptions in the Riemannian setting, we establish locally superlinear, quadratic convergence for the Newton version of RIP and locally linear, superlinear convergence for the quasi-Newton version. These are generalizations of the classical local convergence theory of primal-dual interior point algorithms for nonlinear programming proposed by El-Bakry et al. and Yamashita et al. in 1996.