  Quasiparticle bands of the two-dimensional Hubbard model are calculated using
the Roth two-pole approximation to the one particle Green's function. Excellent
agreement is obtained with recent Monte Carlo calculations, including an
anomalous volume of the Fermi surface near half-filling, which can possibly be
explained in terms of a breakdown of Fermi liquid theory. The calculated bands
are very flat around the (pi,0) points of the Brillouin zone in agreement with
photoemission measurements of cuprate superconductors. With doping there is a
shift in spectral weight from the upper band to the lower band. The Roth method
is extended to deal with superconductivity within a four-pole approximation
allowing electron-hole mixing. It is shown that triplet p-wave pairing never
occurs. Singlet d_{x^2-y^2}-wave pairing is strongly favoured and optimal
doping occurs when the van Hove singularity, corresponding to the flat band
part, lies at the Fermi level. Nearest neighbour antiferromagnetic correlations
play an important role in flattening the bands near the Fermi level and in
favouring superconductivity. However the mechanism for superconductivity is a
local one, in contrast to spin fluctuation exchange models. For reasonable
values of the hopping parameter the transition temperature T_c is in the range
10-100K. The optimum doping delta_c lies between 0.14 and 0.25, depending on
the ratio U/t. The gap equation has a BCS-like form and (2*Delta_{max})/(kT_c)
~ 4.
