  Two-particle lattice states are important for physics of magnetism,
superconducting oxides, and cold quantum gases. The quantum-mechanical lattice
problem is exactly solvable for finite-range interaction potentials. A two-body
Schroedinder equation can be reduced to a system of linear equations whose
numbers scale with the number of interacting sites. For the simplest cases such
as on-site or nearest-neighbor attractions, many pair properties can be derived
analytically, although final expressions can be quite complicated. In this
work, we systematically investigate bound pairs in one-, two-, and
three-dimensional lattices. We derive pairing conditions, plot phase diagrams,
and compute effective masses, radii, and energies. Along the way, we analyze
nontrivial physical effects such as light pairs and the dependence of binding
thresholds on pair momenta. At the end, we discuss the preformed-pair mechanism
of superconductivity and stability of many-pair systems against phase
separation. The paper is a combination of original work and pedagogical
tutorial.
