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Various series expansions have been developed for the two-layer, S=1/2,square lattice Heisenberg antiferromagnet. High temperature expansions are usedto calculate the temperature dependence of the susceptibility and specificheat. At T=0, Ising expansions are used to study the properties of theN\'{e}el-ordered phase, while dimer expansions are used to calculate theground-state properties and excitation spectra of the magnetically disorderedphase. The antiferromagnetic order-disorder transition point is determined tobe $(J_2/J_1)_c=2.537(5)$. Quantities computed include the staggeredmagnetization, the susceptibility, the triplet spin-wave excitation spectra,the spin-wave velocity, and the spin-wave stiffness. We also estimates that theratio of the intra- and inter-layer exchange constants to be $J_2/J_1\simeq0.07$ for cuprate superconductor $YBa_2Cu_3O_{6.2}$.Comment: RevTeX, 9 figure

Various series expansions for the bilayer S= Heisenberg antiferromagnet