  The systematic study of CR manifolds originated in two pioneering 1932 papers
of \'Elie Cartan. In the first, Cartan classifies all homogeneous CR
3-manifolds, the most well-known case of which is a one-parameter family of
left-invariant CR structures on $\mathrm{SU}_2 = S^3$, deforming the standard
`spherical' structure. In this paper, mostly expository, we illustrate and
clarify Cartan's results and methods by providing detailed classification
results in modern language for four 3-dimensional Lie groups. In particular, we
find that $\mathrm{SL}_2(\mathbb{R})$ admits two one-parameter families of
left-invariant CR structures, called the elliptic and hyperbolic families,
characterized by the incidence of the contact distribution with the null cone
of the Killing metric. Low dimensional complex representations of
$\mathrm{SL}_2(\mathbb{R})$ provide CR embedding or immersions of these
structures. The same methods apply to all other three-dimensional Lie groups
and are illustrated by descriptions of the left-invariant CR structures for
$\mathrm{SU}_2$, the Heisenberg group, and the Euclidean group.
