Both the theory of moduli of curves and Teichmüller theory seem to be concerned with the moduli of Riemann surfaces.
However, they appear to belong to different fields within mathematics.
Could someone explain in which ways these approaches to studying Riemann surfaces differ and wherein their similarities lie?
Also, I have seen that in both theories compactifications appear to play an important role and I would be interested to hear if these are somehow related. I have heard that to compactify, one has to add curves of a different genus. Does this apply to both theories?