In this talk we will define braided surfaces with caps, which generalizes the notion of braided surfaces to nonribbon surfaces in D4. We show that any surface in D4 can be isotoped to a braided surface with caps with only positive branch points, and that this isotopy can be taken rel boundary if the boundary is already a classical closed braid in S3. We will then discuss some applications of these surfaces to constructing Lefschetz fibrations with prescribed boundary open book decompositions and braiding closed surfaces in CP2.