Even before the introduction of Conformal Field Theory by Belavin, Polyakov and Zamolodchikov, it appeared indirectly in the work of den Nijs and Nienhuis using Coulomb gas techniques. The latter postulate (unrigorously) that height functions of lattice models converge to the Gaussian Free Field, allowing to derive many exponents and dimensions of 2D lattice models. This convergence is in many ways mysterious, in particular it was never formulated in the presence of a boundary, but rather on a torus or a cylinder. We will discuss possible formulations on general domains or Riemann surfaces and their relations to CFT, SLE and conformal invariance of critical lattice models. Interestingly, new objects in complex geometry and potential theory seem to arise.