 Dutch artist M.C. Escher’s elegant pictorial paradoxes are prized by many, not least by philosophers, physicists, and mathematicians. Some of his work, for example Ascending and Descending, relies on optical illusion to depict what is actually an impossible situation. Other works are paradoxical in the broad sense, but not impossible: Relativity depicts a coherent arrangement of objects, albeit an arrangement in which the force of gravity operates in an unfamiliar fashion. (See the  Other Internet Resources  section below for images.) Quantum gravity itself may be like this: an unfamiliar yet coherent arrangement of familiar elements. Or it may be more like Ascending and Descending, an impossible construction which looks sensible in its local details but does not fit together into a coherent whole when using presently existing building materials. If the latter is true, then the construction of a quantum theory of gravity may demand entirely unfamiliar elements. Whatever the final outcome, the situation at present is one of flux, with a great many competing approaches vying for the prize. However, it is also important to note that the prize is not always the same: string theorists seek a unified theory of all four interactions that has the power of explaining such things as the numbers of generations of elementary particles and other previous inexplicable properties. Other approaches are more modest, and seek only to bring general relativity in line with quantum theory, without necessarily invoking the other interactions. Hence, the problem of quantum gravity can mean very different things to different researchers and what constitutes a possible solution to one group might not qualify as such to another.