  We construct a variety of novel localized states with distinct topological
structures in the 3D discrete nonlinear Schr{\"{o}}dinger equation. The states
can be created in Bose-Einstein condensates trapped in strong optical lattices,
and crystals built of microresonators. These new structures, most of which have
no counterparts in lower dimensions, range from purely real patterns of dipole,
quadrupole and octupole types to vortex solutions, such as "diagonal" and
"oblique" vortices, with axes oriented along the respective directions
$(1,1,1)$ and $(1,1,0)$. Vortex "cubes" (stacks of two quasi-planar vortices
with like or opposite polarities) and "diamonds" (discrete skyrmions formed by
two vortices with orthogonal axes) are constructed too. We identify stability
regions of these 3D solutions and compare them with their 2D counterparts, if
any. An explanation for the stability/instability of most solutions is
proposed. The evolution of unstable states is studied as well.
