Hybrid mesons are hadrons with a non-trivial gluonic content. In the case of Hybrids with a heavy quarks and a heavy antiquark, one can take advantage of the fact that heavy quarks move slowly and use the technology of Non-Relativistic Effective Field Theories. I will briefly review how Heavy Quarkonium is described in terms of those, namely in Non-Relativistic QCD (NRQCD) and Potential NRQCD (pNRQCD), and then I will apply these ideas to the case of Heavy Hybrids. I will present QCD based results for the spectrum, and for decays to and mixing with Heavy Quarkonium. The latter require to go systematically beyond the Born-Oppenheimer approximation. Finally I will discuss the possible identifications with some of the so called XYZ charmonium and bottomonium states discovered in the last decade, which do not fit well in the usual potential models.