Rational VOAs are by now quite well understood: their representation
theory is captured by a modular tensor category; their characters define a
vector-valued modular form for $SL(2,Z)$; etc. The class of C2-cofinite (logarithmic)
VOAs is the natural generalisation of rationality, but their theory is still much less clear.
This talk reviews and contributes to this theory. It is joint work with Thomas Creutzig. Non UBC Unreviewed Author affiliation: University of Alberta Faculty