The Cryogenic Underground Observatory for Rare Events (CUORE) experiment projects to construct and operate an array of 1000 cryogenic thermal detectors of TeO₂, of a mass of 760 g each, to investigate rare events physics, in particular, double beta decay and non-baryonic particle dark matter. A first step towards CUORE is CUORICINO, an array of 62 bolometers, currently being installed in the Gran Sasso Laboratory. In this paper we report the physics potential of both stages of the experiment regarding neutrinoless double beta decay of ¹³⁰Te, WIMP searches and solar axions.