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.