Perfect codes in the Hamming metric over prime power alphabets were completely classified in the 1970s. For more general alphabets, analogous classifications were obtained for such codes correcting at least three errors. The case of 2-error correcting codes over non-prime power alphabets is apparently rather more difficult to handle. In this talk, I will discuss recent work attacking this problem via a variety of results from Diophantine approximation, both effective and ineffective.