The purpose of this paper is to discuss the (uniform) asymptotic stability of a solution of functional differential equations with infinite delay by using the method of Lyapunov-Razumikhin type. To do this, we shall take the following approach: First we decompose a given equation with infinite delay as a sum of an equation with finite delay and the remainders; next we obtain some perturbation theorems for the equation with finite delay in terms of the arguments of Lyapunov- Razumikhin type; finally we discuss the stability of a solution of the original equation with infinite delay. In particular, this approach is useful for the analysis of integro-differential equations. Indeed, we shall consider some integro-differential equations and obtain some results on the stability properties of a solution.