The main purpose of this chapter is to present a unified treatment of the local theory of NFDE with infinite delay on a class of comparatively comprehensive phase spaces which contain admissible phase spaces described in the literature and BC-the space of bounded continuous functions which are convenient and useful for the investigation of several Volterra integro-differential equations and integral equations. Conditions and assumptions are imposed on functionals defining the equation, and therefore independent of the structure and properties of phase spaces. This permits us to find phase spaces and sufficient conditions guaranteeing existence and uniqueness of solutions according to the property of the equation in question rather than to preassign a phase space to dictate the conditions.