  Based on an unconventional Fermi liquid model, we present several results on
the optimally doped and overdoped cuprate superconductors. For the normal
state, we provide an analytic demonstration, backed by self-consistent
Baym-Kadanoff (BK) numerical calculations, of the linear in $T$ resistivity and
linear in 1/$\epsilon$ optical conductivity, provided the interacting Fermi
liquid has strong peaks in its density of states (van-Hove singularities in 2
dimensions) near the chemical potential $\mu$. Moreover, we find that the
interactions tend to pin these strong density of states peaks close to $\mu$.
We show that the low energy dependence of $\chi_{MMP}$ has a fermionic origin.
We obtain particularly high transition temperatures $T_c$ from our
BK-Eliashberg scheme by introducing an ansatz for the fermionic susceptibility
of the carriers. We postulate that the latter is enhanced in an additive manner
due to the weak antiferromagnetic order of the CuO$_2$ planes. We have obtained
a $d_{x^2-y^2}$ gap with $T_c > 120 ^o$K for n.n. hopping $t=250meV$.
