The problem of expressing numbers as the sum of two cubes of non-zero rational numbers has a long history, some results being given by Diophantus. Several mathematicians have since contributed to our knowledge of the problem, in particular in the discovery of numbers which cannot be expressed as the sum of two cubes. Fermat discovered that 1, 18 and 36 belong to this class of numbers, Euler and Legendre demonstrated that so do 3, 4 and 5 and some infinite sets of such numbers were discovered by Sylvester. However, much of interest still remains to be discovered in this area. In this article I hope to both summarise the existing results on this problem and to produce a new class of numbers which cannot be expressed as the sum of two cubes. Most importantly, a general method of tackling the problem will be described which should stimulate further investigation.