Quantum mechanical lattice models with finite-range attraction between particles have applications for unconventional superconductivity and cold atoms in optical lattices. First, arguments in favor of the existence of short-range attractive correlations in high-temperature superconductors are reviewed. Then, a specific two-dimensional model with nearest-neighbor attraction is introduced and solved for two and three fermions. The bound cluster of three fermions with total spin S = 3/2 is found to be Borromean: it exists in the absence of bound pairs. In the S = 1/2 sector, a parameter region is identified where a fermion pair repels a third fermion, which indicates stability of pairs against clustering. The paramagnetic to ferromagnetic (Nagaoka) transition is predicted in the limit of strong attraction. Finally, an anisotropic three-dimensional model is considered. The pair condensation temperature is found to be maximal at an intermediate interlayer hopping. A phase diagram is proposed and its relevance to high-temperature superconductors is discussed.