  Cooper pairing in two dimensions is analyzed with a set of renormalized
equations to determine its binding energy for any fermion number density and
all coupling assuming a generic pairwise residual interfermion interaction. \
Also considered are Cooper pairs (CPs) with nonzero center-of-mass momentum
(CMM)--usually neglected in BCS theory--and their binding energy is expanded
analytically in powers of the CMM up to quadratic terms. A Fermi-sea-dependent
{\it linear} term in the CMM dominates the pair excitation energy in weak
coupling (also called the BCS regime) while the more familiar quadratic term
prevails in strong coupling (the Bose regime). The crossover, though strictly
unrelated to BCS theory {\it per se,} is studied numerically as it is expected
to play a central role in a model of superconductivity as a Bose-Einstein
condensation of CPs where the transition temperature vanishes for all
dimensionality $d\leq 2$ for quadratic dispersion, but is {\it nonzero} for all
$d\geq 1$ for linear dispersion.
