  In the last few years evidence has been accumulating that there are a
multiplicity of energy scales which characterize superconductivity in the
underdoped cuprates. In contrast to the situation in BCS superconductors, the
phase coherence temperature Tc is different from the energy gap onset
temperature T*. In addition, thermodynamic and tunneling spectroscopies have
led to the inference that the order parameter $\Delta_{sc}$ is to be
distinguished from the excitation gap $\Delta$; in this way, pseudogap effects
persist below Tc. It has been argued by many in the community that the presence
of these distinct energy scales demonstrates that the pseudogap is unrelated to
superconductivity. In this paper we show that this inference is incorrect. We
demonstrate that the difference between the order parameter and excitation gap
and the contrasting dependences of T* and Tc on hole concentration $x$ and
magnetic field $H$ follow from a natural generalization of BCS theory. This
simple generalized form is based on a BCS-like ground state, but with self
consistently determined chemical potential in the presence of arbitrary
attractive coupling $g$. We have applied this mean field theory with some
success to tunneling, transport, thermodynamics and magnetic field effects. We
contrast the present approach with the phase fluctuation scenario and discuss
key features which might distinguish our precursor superconductivity picture
from that involving a competing order parameter.
