Geometry of third-order ODEs

We address the problem of local geometry of third order ODEs modulo contact, point and fibre-preserving transformations of variables. Several new and already known geometries are described in a uniform manner by the Cartan method of equivalence. This includes conformal, Weyl and metric geometries in three and six dimensions and contact projective geometry in dimension three. Respective connections for these geometries are given and their curvatures are expressed by contact, point or fibre-preserving relative invariants of the ODEs.