Asymptotic mapping class group of genus zero was first introduced by Funar and Kapoudjian. It is a finitely presented group which contains the mapping class groups of all genus zero surfaces. Later the definitions were generalized to allow the genus to be any finite number (Funar--Aramayona) and infinite (Funar--Kapoudjian). We will discuss how these groups are constructed and show that they are in fact of type F_infinity. The proof boils down to prove certain subsurface complexes are highly connected. This is based on joint work with Javier Aramayona, Kai-Uwe Bux, Jonas Flechsig and Nansen Petrosyan.