In this paper, we introduce the class of n-quasi-m-isometric op-erators on a Banach space. Such a class seems to be a natural generalizationof m-isometric operators on Banach spaces and of n-quasi-m-isometric oper-ators on Hilbert spaces. We start by giving some of their elementary prop-erties, studying the products and the power of such operators. Next, we willexplain the relation between C 0 -semigroups, n-quasi-(m, q)-isometric operatorsand (m, q)-isometric operators. Finally, we focus on the dynamic of an n-quasi-m-isometry. More precisely, we prove a result characterizing the supercyclicityof such a class.