Biharmonic maps are generalizations of harmonic maps and biharmonic functions. As solutions of a system of 4th order PDEs, examples and the general properties of biharmonic maps are hard to reveal. In this talk, we will talk about some recent work on the study of biharmonic maps among conformal maps. These include examples and classifications of biharmonic conformal immersions of surfaces, biharmonic conformal maps between manifolds of the same dimension, and the links between conformal biharmonicity and the notion of $f$-biharmonic maps and the equations of Yamabe type.