Abstract We resolve the isomorphism problem for tensor algebras of unital multivariable dynamical systems. Specifically, we show that unitary equivalence after a conjugation systems is complete invariant isometric isomorphisms between their algebras. In particular, this settles conjecture Davidson and Kakariadis, Inter. Math. Res. Not. 2014 (2014), 1289–1311 relating to work Arveson, Acta 118 (...