We study rational maps from a projective space of dimension two to another of dimension three, both over the same field. We will start by giving the general framework and first results obtained on this question by Botbol, Busé and myself. Then I will turn to questions concerning the fibers of dimension one that such a map can have and present two way to address this question, the first by Tran Quang Hoa, and the second by the same author together with Dale Cutkosky and myself. Examples show that our estimates are pretty sharp, but leave possibilities for improvement.