In this paper we present an arbitrary-order fully discrete Stokes complex on general polygonal meshes. Based upon the recent construction of the de Rham fully discrete complex by D. A. Di Pietro and J. Droniou we extend it using the same principle. We complete it with other polynomial spaces related to vector calculus operators and to the Koszul complex required to accommodate the increased smoothness of the Stokes complex. This complex is especially well suited for problem involving Jacobian, divergence and curl, like e.g. the Stokes system or magnetohydrodynamics. We show a complete set of results on the novelties of this complex complementing those of D. A. Di Pietro and J. Droniou: exactness properties, uniform Poincaré inequalities and primal and adjoint consistency. We use our new complex on the Stokes system and validate the expected convergence rates with various numerical tests.