The group of plane Cremona transformations is endowed with a natural topology and the set of maps of some bounded degree is closed. We will review some known properties of the quasi-projective variety which parametrizes plane The group of plane Cremona transformations is endowed with a natural topology and the set of maps of some bounded degree is closed. We will review some known properties of the quasi-projective variety which parametrizes plane Cremona maps of fixed degree. Then, we will address the question of determining which plane Cremona maps of small degree are limits of maps of higher degree. This talk is mainly based on joint works with Jérémy Blanc and with Cinzia Bisi and Massimiliano Mella.