Suppose that $(M,E)$ is a compact contact manifold, and that a compact Lie
group $G$ acts on $M$ transverse to the contact distribution $E$. In an earlier
paper, we defined a $G$-transversally elliptic Dirac operator $\dirac$,
constructed using a Hermitian metric $h$ and connection $\nabla$ on the
symplectic vector bundle $E\rightarrow M$, whose equivariant index is
well-defined as a generalized function on $G$, and gave a formula for its
index. By analogy with the geometric quantization of symplectic manifolds, the
$\mathbb{Z}_2$-graded Hilbert space $Q(M)=\ker \dirac \oplus \ker \dirac^{*}$
can be interpreted as the "quantization" of the contact manifold $(M,E)$; the
character of the corresponding virtual $G$-representation is then given by the
equivariant index of $\dirac$. By defining contact analogues of the algebra of
observables, pre-quantum line bundle and polarization, we further extend the
analogy by giving a contact version of the Kostant-Souriau approach to
quantization, and discussing the extent to which this approach is reproduced by
the index-theoretic method.