Some of the material of this paper is based on joint unpublished work of the author and Peter Jones. For a fixed atomic measure with compact support in the complex plane, a variety of spectra can be defined. These include the box dimension spectrum \(f(\alpha)\), the packing spectrum \(\pi(t)\), the Hausdorff dimension spectrum, and others. This survey reviews results on the singularities of harmonic measure, and how it is distributed over the boundary. In this situation, a dimension spectrum characterizes the size of the sets where, say, \(\omega_\alpha(B(z,\delta))\approx\delta^\alpha\) for small \(\delta\). The first two sections are devoted to multifractal formalism and harmonic measure on regular fractals. When \(\omega\) is the harmonic measure on a conformal Cantor set, the Hausdorff and box dimension spectra coincide, as do the packing and covering spectra. These results carry over to the situation of regular fractals where \(\partial\Omega\) is a mized repeller with respect to an analytic dynamics. If \(\Omega\) is a simply connected domain with compact boundary, another spectrum is available, the integral means spectrum \(\beta(t)\) related to the derivative of the Riemann map. It is shown that \(\beta(t)= \pi(t)+t-1\) for \(t\leq t*\), where \(t*\) is the first zero of \(\pi(t)\). Universal spectra \(F(\alpha)\) and \(\Pi(t)\) are introduced next. These are the suprema of certain functions related to \(f(\alpha)\) and \(\pi(t)\), the suprema being taken over all plane domains with compact boundaries. \(F(\alpha)\) and \(\Pi(t)\) are shown to satisfy Legendre type relations. Similar results hold when \(\Omega\) ranges over the simply connected domains, or over the domains whose boundaries are Cantor sets. The method of fractal approximation is applied to problems concerning the coefficients and integral means of univalent functions. Finally, multifractal analysis is exhibited for the harmonic measures associated with polynomial Julia sets and random snowflakes.