In 1985, Catlin developed a powerful reduction technique to study the existence of spanning and dominating Eulerian subgraphs. He discovered a family of graphs, called collapsible graph, which plays an important role in his reduction technique. He then applied the reduction method to the study of integer flows, double cycle covers, edge-connectivity, Hamiltonian line graphs, among others. He had posed several conjectures related to collapsible graphs and Eulerian subgraphs. In this talk, we will discuss some new developments on the study of Eulerian subgraphs and related topics, and some of Catlin's conjectures.