In this paper we investigate the generalized Hyers-Ulam stability of mappings of m-semigroups m ∊ N; m ≥ 2 into Banach spaces. For m = 3 the results can be found in Amyari and Moslehian [Approximate homomorphisms of ternary semigroups, Lett. Math. Phys. 77 (2006), 1-9] with the mention that they are true in the class of normal m-semigroups which is larger than the class of commutative m-semigroups. For m = 2 we find certain results of Hyers [On the stability of the linear functional equation, Proc. Nat. Acad. Sci. USA, 27 (1941), 222-224], Aoki [On the stability of the linear transformation in Banach spaces,J. Math. Soc. Japan, 2 (1950), 64-66], Rassias, Th. M. [On the stability of the linear mapping in Banach space, Proc. Amer. Math, Soc. 72 (1978), 297-300 ] and Rassias, J. M. [Solution of a Problem of Ulam, J. Approx. Theory Math. 57 (1989), 268-273]. In addition, we establish the superstability of m-ary homomorphims into Banach algebras endowed with multiplicative norms, generalizing the results of Szekelyhidi [On a theorem of Baker, Lawrence and Zorzitto, Proc. Amer. Math. Soc., 84 (1982), 95-96] and Amyari and Moslehian (2006). [ABSTRACT FROM AUTHOR]