We first quickly review some basic facts about quantum theories in 1+5 dimensions with 2,0 superconformal symmetry. They arise as decoupled subsectors of string or M theories on space times containing certain defects like branes or singularities. By compactification, they are also related to Yang-Mills theories in lower dimensions. We then describe, in somewhat more detail, recent work on these theories considered on a spatial five torus. In particular, one may determine the spectrum of vacuum states, and also learn much about the spectrum of BPS states. Hopefully, this will eventually be helpful for giving a definition of these mysterious theories.