A bstract The dark matter may consist not of one elementary particle but of different species, each of them contributing a fraction of the observed dark matter density. A major theoretical difficulty with this scenario — dubbed multi-component dark matter — is to explain the stability of these distinct particles. Imposing a single Z N symmetry, which may be a remnant of a spontaneously broken U(1) gauge symmetry, seems to be the simplest way to simultaneously stabilize several dark matter particles. In this paper we systematically study scenarios for multi-component dark matter based on various Z N symmetries ( N ≤ 10) and with different sets of scalar fields charged under it. A generic feature of these scenarios is that the number of stable particles is not determined by the Lagrangian but depends on the relations among the masses of the different fields charged under the Z N symmetry. We explicitly obtain and illustrate the regions of parameter space that are consistent with up to five dark matter particles. For N odd, all these particles turn out to be complex, whereas for N even one of them may be real. Within this framework, many new models for multi-component dark matter can be implemented.