This paper presents and analyzes a new multigrid framework to solve shape optimization problems governed by elliptic PDEs. The boundary of the domain, i.e., the control variable, is represented as the graph of a continuous function that is approximated at various levels of discretization. The proposed multigrid shape optimization scheme acts directly on the function describing the geometry of the domain and it combines a single-grid shape gradient optimizer with a coarse-grid correction (minimization) step, recursively within a hierarchy of levels. The convergence of the proposed multigrid shape optimization method is proved and several numerical experiments assess its effectiveness.