A flat plane can be tiled, for example, with regular hexagons. If you try to tile a sphere with hexagons, however, it doesn't work--you have to introduce 12 pentagons to complete the tiling. Try to tile a sphere with triangles, and you can do it, but there will still be 12 defects where you have 5 triangles surrounding a vertex rather than the regular 6.
Meanwhile, flat 3D space can be tiled, for example, with truncated octahedra, or disphenoid tetrahedra. But what happens if we try to tile a 3-sphere? Is it known what kinds of defects will be introduced?
(I presume the answer will be related to regular 4D polytopes, the way that the 12-point defects in 2-sphere tilings are related to dodecahedra, but I have no idea how to convert that intuition into a proof of what the shapes around the defect points might be for different possible face tilings.)