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We explore the concept that the above-T c state is a normal Fermi liquid with strong correlations of the type found in valence-fluctuation and heavy-fermion materials. Experimental evidence and theoretical arguments for this phase are presented. Using an Anderson lattice model Hamiltonian, and a variational many-body formalism, the finite-U mechanism for s-like (full point-group symmetry) pairing is examined in detail. At the mean field or (1/N)0 level of approximation, the pairing tendency is strongly opposed by a magnetic tendency arising from a Gutzwiller version of hybridization renormalization. Pairing does not seem possible for realistic Hamiltonian parameters. This finite-U treatment has, however, now been extended to the (1/N)1 level of approximation. The lattice aspect is found to play a major and unexpected role here, leading to strong suppression of the magnetic tendency. This refined treatment is now found to provide adequate pairing attraction, for reasonable Hamiltonian parameters. Superconductivity is found only when the charge transfer energy Δ CT is quite large, Δ CT ≳ U, but there is evidence that Δ CT is indeed of this magnitude in the cuprate materials. Adequate band narrowing (mass enhancement or heaviness) and a very short coherence length are also obtained, in reasonable agreement with experiment. The quasiparticle interaction at the Fermi surface is strongly repulsive. This provides a reasonable source for the large and linear resistivity, as well as for other aspects of marginal-Fermi-liquid phenomenology. Evidence is found for a strong reduction of T c due to pair-breaking by the quasiparticle scattering. The main (fully self-consistent) calculations assume in-plane isotropy. However, a simplified calculation with a more realistic (anisotropic) band structure for the CuO 2 plane is found to provide a highly anisotropic gap, which may or may not have nodes. There is considerable evidence for such types of gap structure. Because a connection is found here between anomalously short coherence length and anomalously high resistivity, we suggest that this mechanism is also operating within other families of "exotic" superconductors.