  A PhD thesis written under supervision of Pawel Nurowski and defended at the
Faculty of Physics of the University of Warsaw.
  We adress the problems of local equivalence and 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.
  We construct Cartan coframes which yield the full set of local invariants and
solve the local problem of contact and point equivalence of the ODEs. We
explicitly describe ODEs admitting at least four-dimensional Lie group of
contact or point symmetries and real ODEs fibre-preserving equivalent to II,
IV, V, VI, VII and XI Chazy classes.
