In the world around us we are used to seeing tiled structures from patterns on clothes to floor coverings and artwork. Typically these structures are highly ordered and comprise symmetric arrays of their components. Our familiarity with such arrangements can lead us to overlook random tiling and quasi-crystalline structures that are present all around us. Indeed such structures are found in a broad spectrum of environments from glasses to Islamic art. We recently demonstrated that it is possible to create random tiled structures using molecules with carefully designed dimensions and intermolecular interactions. Such molecules can be considered as simple tiles and our original work has opened the possibility of studying the self-assembly of structures that are designed to avoid translational order. Our studies move beyond the conventional paradigms of supramolecular chemistry and are more akin to the behaviour of natural systems. The construction of random tiling systems, such as those that will be prepared, has great importance to a variety of disciplines, moving beyond chemistry and physics to scientists working on optimisation problems and statistical mechanics, to those researching quasi-crystals and the so-called spin-ice problem.