Mapping classes of surfaces of finite type have been classified by Nielsen and Thurston. For surfaces of infinite type (e.g. surfaces of infinite genus), no such classification is known. I will talk about the difficulties that arise when trying to generalize the Nielsen-Thurston classification to infinite-type surfaces and present a first result in this direction, concerning maps which - loosely speaking - do not show any pseudo-Anosov behavior. Joint work with Mladen Bestvina and Jing Tao.[-]