Given a group, we are interested in understanding and classifying its  
actions on one-dimensional manifolds, that its representations into  
the groups of homeomorphisms or diffeomorphisms of an interval or the  
circle. In this talk we will address this problem for a class of  
groups arising via an action on intervals of a special type, called  
locally moving. A well studied example in this class is the Thompson  
group. We will see that if G is a locally moving group of  
homeomorphisms of a real interval, then every action of G on an  
interval by diffeomorphisms (of class C^1) is semiconjugate to the  
natural defining action of G. In contrast such a group can admit a  
much richer space of actions on intervals by homeomorphisms, and we  
will investigate the structure of such actions. This is joint work  
with Joaquín Brum, Cristóbal Rivas and Michele Triestino.