I will give an introduction to a class of geometric structures known as
parabolic geometries: these are Cartan geometries modelled on
homogeneous spaces of the form $G/P$, where $G$ is a semisimple Lie group
and $P$ is a parabolic subgroup. The most prominent example of a parabolic
geometry is conformal geometry in dimension $>2$; the symmetry group $G$
of the flat homogeneous model in this case is the conformal group. A more
exotic but still classical example is the geometry of $(2,3,5)$
distributions, which is related to the exceptional simple Lie group
$G=G_2$. In this talk I will review some history, explain how the Lie
group $G_2$ appears in this context, and discuss recent developments in
the field.