Energy minimizing harmonic maps from a three-ball into the two-sphere are well understood. A natural existence question for a continuous harmonic map with a suitably given Dirichlet boundary value or in a given homotopic class [a problem that I learned from R.Schoen in mid-80s] remains open. We shall review several earlier results including the notions of the relaxed energies and minimal connections. The main goal of this lecture is to describe related progress in higher dimensions. In particular, a proof of the relaxed energy formula proposed by H. Brezis and P. Mironescu will be explained.