  We address the properties of fully three-dimensional solitons in complex
parity-time (PT)-symmetric periodic lattices with focusing Kerr nonlinearity,
and uncover that such lattices can stabilize both, fundamental and
vortex-carrying soliton states. The imaginary part of the lattice induces
internal currents in the solitons that strongly affect their domains of
existence and stability. The domain of stability for fundamental solitons can
extend nearly up to the PT-symmetry breaking point, where the linear lattice
spectrum becomes complex. Vortex solitons feature spatially asymmetric profiles
in the PT-symmetric lattices, but they are found to still exist as stable
states within narrow regions. Our results provide the first example of
continuous families of stable three-dimensional propagating solitons supported
by complex potentials.
