After a brief introduction, we consider three main results in the existence theory of harmonic maps between manifolds. The first is the heat-equation proof of Eells and Sampson, which says that minimal harmonic maps of compact manifolds into compact manifolds with nonpositive curvature always exist. The next two results show they exist among maps of compact Riemann surfaces into compact manifolds, N, with $ pi sb2$(N) = 0. One proof uses the induced $ pi sb1$-action of Schoen and Yau; the other a perturbation of the action due to Sacks and Uhlenbeck. As required, we also develop some of the regularity theory, especially that for surfaces.

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