The space of hyperbolic metrics on a (marked) hyperbolic surface was studied by Teichmueller around 1940. An alternative viewpoint is via discrete and faithful representations to SL(2,R) of the fundamental group. Later the space of discrete and faithful (quasi-Fuchsian) representations of a surface group to SL(2,C) were considered. More recently, the space of representations to SU(2,1) has been a subject of much research. I will briefly outline the two classical sitautions and then give a survey of known work in the case of SU(2,1).