  We introduce a new perspective on the classical Nirenberg problem of
understanding the possible Gauss curvatures of metrics on $S^{2}$ conformal to
the round metric. A key tool is to employ the smooth Cheeger-Gromov compactness
theorem to obtain general and essentially sharp a priori estimates for Gauss
curvatures $K$ contained in naturally defined stable regions. We prove that in
such stable regions, the map $u \rightarrow K_{g}$, $g = e^{2u}g_{+1}$ is a
proper Fredholm map with well-defined degree on each component. This leads to a
number of new existence and non-existence results. We also present a new proof
and generalization of the Moser theorem on Gauss curvatures of even conformal
metrics on $S^{2}$.
  In contrast to previous work, the work here does not use any of the
Sobolev-type inequalities of Trudinger-Moser-Aubin-Onofri.
