  We provide five examples of conformal geometries which are naturally
associated with ordinary differential equations (ODEs). The first example
describes a one-to-one correspondence between the Wuenschmann class of 3rd
order ODEs considered modulo contact transformations of variables and (local)
3-dimensional conformal Lorentzian geometries. The second example shows that
every point equivalent class of 3rd order ODEs satisfying the Wuenschmann and
the Cartan conditions define a 3-dimensional Lorentzian Einstein-Weyl geometry.
The third example associates to each point equivalence class of 3rd order ODEs
a 6-dimensional conformal geometry of neutral signature. The fourth example
exhibits the one-to-one correspondence between point equivalent classes of 2nd
order ODEs and 4-dimensional conformal Fefferman-like metrics of neutral
signature. The fifth example shows the correspondence between undetermined ODEs
of the Monge type and conformal geometries of signature $(3,2)$. The Cartan
normal conformal connection for these geometries is reducible to the Cartan
connection with values in the Lie algebra of the noncompact form of the
exceptional group $G_2$. All the examples are deeply rooted in Elie Cartan's
works on exterior differential systems.
