Harmonic maps are critical points of the energy functional in between manifolds.
Existence and regularity properties of the map have been a subject of intensive study in geometric analysis. A remarkable result of Helein in 1990 establishes that any weak harmonic map defined on a compact surface is already smooth. In this talk, I will survey results in the field, give a simplier proof of the result of Helein when the target manifold of the map is spheres and indicate the extension of the regularity results to bi-harmonic maps. The method proves useful in a variety of asymptotic problems of coding and information theory. A common difficulty in the study of turbulent transport is the absence of any separation between the scale of the energy containing eddies and the scale of the inhomogeneity of the turbulence.