In this paper, we analyze metrical approximations of functions F : Λ × X → Y by trigonometric polynomials and ρ-periodic type functions, where ∅ = Λ ⊆ R n , X and Y are complex Banach spaces, and ρ is a general binary relation on Y . Besides the classical concept, we analyze Stepanov, Weyl, Besicovitch and Doss generalized approaches to metrical approximations. We clarify many structural properties of introduced spaces of functions and provide several applications of our theoretical results to the abstract Volterra integrodifferential equations and the partial differential equations.2010 Mathematics Subject Classification. 42A75, 43A60, 47D99. Key words and phrases. Metrical approximations of functions by trigonometric polynomials, metrical approximations of functions by ρ-periodic type functions, abstract Volterra integrodifferential equations.