<p>The aim of this paper is to study <i>m</i>-isometric weighted shifts with operator weights (both unilateral and bilateral). We obtain a characterization of such shifts by polynomials with operator coefficients. The procedure of construction of all <i>m</i>-isometric weighted shifts with positive weights is presented. We answer the question when the weights of <i>m</i>-isometric shift are commuting. The completion problem for <i>m</i>-isometric weighted shifts with operator weights is solved. We characterize <i>m</i>-isometric bilateral shifts the adjoints of which are also <i>m</i>-isometric.</p>