The main purpose of this paper is to introduce the class of (equi-) Weyl $C^{(n)}$-almost periodic functions and the class of asymptotically (equi-) Weyl $C^{(n)}$-almost periodic functions as well as to examine the existence and uniqueness of solutions of abstract inhomogenous Volterra integro-differential equations belonging these classes of functions. The Besicovitch-Doss $C^{(n)}$-almost periodic functions and solutions of abstract inhomogenous Volterra integro-differential equations belonging this class of functions are also considered.