Let n∈ℕ and let A and B be rings. An additive map h:A→B is called an n-Jordan homomorphism if h(an)=(h(a))n for all a∈A. Every Jordan homomorphism is an n-Jordan homomorphism, for all n≥2, but the converse is false in general. In this paper we investigate the n-Jordan homomorphisms on Banach algebras. Some results related to continuity are given as well.