Biharmonic maps are natural generalizations of harmonic maps. The fact that the Euler equations are fourth-order (as opposed to second order for harmonic maps) presents a number of analytic challenges. In this talk, I will discuss biharmonic maps with S^1-equivariant symmetry and the associated biharmonic heat flow.  In particular, I will present a recent uniqueness result for S^1-equivariant biharmonic maps from R^4 into S^4. I will further discuss the formation of singularities in the heat flow.