Tiling is the arrangement of

geometrical shapes such that they can cover an entire

plane, or so that they fill some other shape, without any gaps or overlap.

Most tilings of the plane are periodic, that is, the pattern repeats itself endlessly across the plane, but Roger Penrose discovered a pair of shapes which can only tile the plane non-periodically.

Besides just searching for tilings with given properties, there is a class of problems related to filling a given space with other given tiles that also falls under the general category of tiling. Among these is the famous squared square problem. Many tiling problems have elegant methods of solution (or proofs of impossibility) due to parity problems.