In quantum physics, and more specifically in quantum optics, several notions of classical and hence nonclassical state are in use. They rely on the positivity of quasi-probability distributions, specifically the Glauber-Sudarshan or Wigner functions of the state. Characterizing the classical states is in general a difficult task, involving interesting questions of functional and harmonic analyss. In this talk, we will, after reviewing the subject, report on some recent progress.