The aim of this paper is to introduce n-ary BiHom-algebras, generalizing BiHom-algebras. We introduce an alternative concept of BiHom-Lie algebra called BiHom-Lie-Leibniz algebra and study various type of n-ary BiHom-Lie algebras and BiHom-associative algebras. We show that n-ary BiHom-Lie-Leibniz algebra can be represented by BiHom-Lie-Leibniz algebra through fundamental objects. Moreover, we provide some key constructions and study n-ary BiHom-Lie algebras induced by (n-1)-ary BiHom-Lie algebra.