Abstract: Cone structures on smooth (contact) manifolds are defined as one-dimensional subbundles of Grassmann (resp., isotropic Grassmann) bundles. We define the notions of regularity, type and algebraic symbol for these structures and prove the existence of the canonical frame. Further, we discuss two main examples of these structures coming from the geometry of systems of ODEs of mixed order and the local equivalence problem of bracket-generating vector distributions of maximal class.