  We establish theorems on the existence and compactness of solutions to the
$\sigma_2$-Nirenberg problem on the standard sphere $\mathbb S^2$. A first
significant ingredient, a Liouville type theorem for the associated fully
nonlinear M\"obius invariant elliptic equations, was established in an earlier
paper of ours. Our proof of the existence and compactness results requires a
number of additional crucial ingredients which we prove in this paper: A
Liouville type theorem for the associated fully nonlinear M\"obius invariant
degenerate elliptic equations, a priori estimates of first and second order
derivatives of solutions to the $\sigma_2$-Nirenberg problem, and a B\^ocher
type theorem for the associated fully nonlinear M\"obius invariant elliptic
equations. Given these results, we are able to complete a fine analysis of a
sequence of blow-up solutions to the $\sigma_2$-Nirenberg problem. In
particular, we prove that there can be at most one blow-up point for such a
blow-up sequence of solutions. This, together with a Kazdan-Warner type
identity, allows us to prove $L^\infty$ a priori estimates for solutions of the
$\sigma_2$-Nirenberg problem under some simple generic hypothesis. The higher
derivative estimates then follow from classical estimates of Nirenberg and
Schauder. In turn, the existence of solutions to the $\sigma_2$-Nirenberg
problem is obtained by an application of the by now standard degree theory for
second order fully nonlinear elliptic operators.
