Biharmonic maps are maps between Riemannian manifolds that are critical points of the bienergy functional, they are solutions of a system of 4thorder PDEs. Biharmonic maps include harmonic maps, biharmonic functions and biharmonic submenisolds as special examples. The talk will be focused on biharmonic conformal immersions, biharmonic conformal submersions and their relations to the maps between manifolds that preserve solutions of bi-Laplace equations, and biharmonic conformal maps between manifolds of the same dimension and their links to Yamabe-type equations.