Modern naturally occurring proteins have been produced by a lengthy selective evolutionary process. While, in general, they are all composed of the same 20 amino acids there is a distinct bias in their average amino acid composition. This bias may have arisen due to evolutionary mechanisms, the degeneracy of the genetic code, the primordial availability of suitable monomers, their relative reactivity or a number of other, equally speculative, causes. Mathematics appears to dictate that Nature could not have sampled all possible amino acid sequences and selected the most suitable for a particular function, suggesting that the proteins observed today may have evolved from a relatively small number of precursors. If this is true it would imply that there is a vast set of possible proteins that have simply never existed and that may possess interesting or useful properties. This article investigates whether the structural space occupied by proteins that do not currently exist can be sampled. One approach suggests itself – random polypeptide synthesis in which all possible residue types are inserted at all possible positions of an amino acid sequence of a given length. It is abundantly clear that the truly random synthesis of even a small set of such protein sequences is precluded by simple mathematics. The issues that this raises are discussed and different practical approaches to the problem described.