  We study the fate of superconductivity in the vicinity of a class of metallic
quantum critical points obtained by coupling a Fermi surface to a critical
boson. In such systems there is a competition between the enhanced pairing
tendency due to the presence of long-range attractive interactions near
criticality, and the suppression of superconductivity due to the destruction of
the Landau quasiparticles. We show that there are regimes in which these two
effects offset one another, resulting in a novel non-Fermi liquid fixed point
with finite, scale invariant, BCS coupling. While these interactions lead to
substantial superconducting fluctuations, they do not drive the system into a
superconducting ground state. The metallic quantum critical fixed points are
connected to the superconducting regime by a continuous phase transition. These
results are established using a controlled expansion in the deviation from d=3
spatial dimensions, as well as in a large number N of internal flavors. We
discuss the possible relevance of our findings to the phenomenon of
superconducting domes condensing out of a non-Fermi liquid normal state near
quantum critical points.
