  Since the discovery of high-temperature superconductivity in 1986 by Bednorz
and Mueller, great efforts have been devoted to finding out how and why it
works. From the d-wave symmetry of the order parameter, the importance of
antiferromagnetic fluctuations, and the presence of a mysterious pseudogap
phase close to the Mott state, one can conclude that high-Tc superconductors
are clearly distinguishable from the well-understood BCS superconductors. The
d-wave superconducting state can be understood through a Gutzwiller-type
projected BCS wave-function. In this review article, we revisit the Hubbard
model at half-filling and focus on the emergence of exotic superconductivity
with d-wave symmetry in the vicinity of the Mott state, starting from ladder
systems and then studying the dimensional crossovers to higher dimensions. This
allows to confirm that short-range antiferromagnetic fluctuations can mediate
superconductivity with d-wave symmetry. Ladders are also nice prototype systems
allowing to demonstrate the truncation of the Fermi surface and the emergence
of a Resonating Valence Bond (RVB) state with preformed pairs in the vicinity
of the Mott state. In two dimensions, a similar scenario emerges from
renormalization group arguments. We also discuss theoretical predictions for
the d-wave superconducting phase as well as the pseudogap phase, and address
the crossover to the overdoped regime. Finally, cold atomic systems with
tunable parameters also provide a complementary insight into this outstanding
problem.
