  In this paper we address the behavior of the superfluid transition
temperature $T_c$ in the attractive Hubbard model. We study systematically the
effects of pairing fluctuations and address all filling fractions over the
entire range of attractive interaction strength. While the attractive Hubbard
model can be regarded as the generalization of BCS to Bose Einstein
condensation (BEC) crossover to a lattice, we find that the BEC limit of this
Hubbard model is very different from that of jellium, owing to the strong
inter-site repulsion between pairs, which becomes important near half filling
when the on-site attraction is strong. A central conclusion of our work is that
in a lattice, around half filling, the smooth evolution from the BCS to the BEC
limits is interrupted. For the attractive Hubbard model, $T_c$ vanishes when
the system approaches the bosonic regime with increasing interaction strength.
We suggest that the vanishing of $T_c$ at strong coupling strength may signal a
quantum critical transition to another form of superfluid not continuously
connected to a BCS-like phase. We present a simple variational ansatz for the
ground state in this more strongly coupled superfluid. We further generalize
the (s-wave) Hubbard model to d-wave pairing and address issues of potential
relevance to high temperature superconductors. For the d-wave case, we present
a phase diagram and show that here too, one observes a vanishing of $T_c$ when
the pairing onset temperature $T^*$ becomes sufficiently large. We suggest that
future experiments on ultracold fermions in optical lattices should not be
exclusively limited to the repulsive Hubbard model, but should address the
attractive model in order to elucidate features of high temperature
superconductivity.
