In this thesis, we explore the concept of Wang tiles, which are polygons with colours on their edges. A set of such tiles can be used to tile the plane by placing the tiles side by side such that corresponding edges have the same colour. Rotations and reflections are not allowed. We discuss the history of Wang tiles and examine two methods for carrying out computations with them. One method simulates a Turing machine and the other uses ‘signals’. We present multiple new tile sets of square, hexagonal, and octagonal Wang tiles which are used to perform computations. To the best of our knowledge, these are the first examples of octagonal Wang tile sets and dihedral Wang tile sets. Additionally, we present two proofs establishing that our new square tile sets for addition and computing the Fibonacci sequence uniquely tile the plane. Furthermore, we disprove previous claims of unique tilings for three square tile sets used for addition, computing the Fibonacci sequence, and computing the prime numbers.