  The multi-layer $S={1\over 2}$ square lattice Heisenberg antiferromagnet with
up to 6 layers is studied via various series expansions. For the systems with
an odd number of coupled planes, the ground-state energy, staggered
magnetization, and triplet excitation spectra are calculated via two different
Ising expansions. The systems are found to have long range N\'eel order and
gapless excitations for all ratios of interlayer to intralayer couplings, as
for the single-layer system. For the systems with an even number of coupled
planes, there is a second order transition point separating the gapless Ne\'el
phase and gapped quantum disordered spin liquid phase, and the critical points
are located via expansions in the interlayer exchange coupling. This transition
point is found to vary about inversely as the number of layers. The triplet
excitation spectra are also computed, and at the critical point the normalized
spectra appear to follow a universal function, independent of number of layers.
