Biharmonic maps are maps between Riemannian manifolds that minimize the bi-energy functional. Biharmonic sumanifolds are those submanifolds whose defining isometric immersions are biharmonic maps. Biharmonic maps generalize the notion of harmonic maps and biharmonic submanifolds include minimal submanifolds as a subclass. In this talk we will review some basic examples of biharmonic maps, some fundamental problems in the study of biharmonic maps, and then we will focus on Chen’s conjecture and the generalized Chen’s conjecture on biharmonic submanifolds. This includes our counter examples to show that the generalized Chen’s conjecture is false.