  We consider Thompson's groups from the perspective of mapping class groups of
surfaces of infinite type. This point of view leads us to the braided Thompson
groups, which are extensions of Thompson's groups by infinite (spherical) braid
groups. We will outline the main features of these groups and some applications
to the quantization of Teichm\"uller spaces. The chapter provides an
introduction to the subject with an emphasis on some of the authors results.
