An open problem (as far as I know) in three-dimensional Game of Life is to construct a “glider gun” for a reasonable three-dimensional ruleset.
One natural set of rules are 5,6,7/6, in which cells stay alive with 5–7 neighbors and are created with 6 neighbors. In this set of rules, some two-dimensional patterns from the original game of life still work if they are copied into two adjacent layers.
Another feature is the “time-space barrier.” No new cells can be created close to its surface. To make it stable, we can wrap it into a cube.
Unfortunately, the “Gosper Glider Gun” from standard Game of Life does not work directly in 3D. It interacts into the third dimension. But if we wedge it between two barrier cubes it works for a while.
Everything unravels when the glider reaches the edge of the cubes. I don’t know whether one can construct something that prevents this from happening.
See also: Bays, Carter (1987), “Candidates for the Game of Life in Three Dimensions”, Complex Systems 3 (1): 373–400.