The self-validating numerical method is sueveyed for nonlinear problems. By taking into account of the effect of rounding error rigorously, this method provides a method of computer assisted proofs. In the first place, Kantrovich's approach to this problem is surveyed. His method is based on his convergence theorem of Newton's method and can be seen as an a posteriori error estimation method. Then, Urabe's approach to this problem is discussed. He treated practical nonlinear differential equations such as the van der Pol equation and the Duffing equation and proved the existence of their periodic and quasi-periodic solutions by the self-validating numerics. Generalizations and abstraction of Urabe's method to more general functional equations is also discussed. Then methods for rigorous estimation of rounding errors are surveyed.