A number of mathematical techniques are presented which have proven successful in obtaining analytic solutions to the differential equations for the dyadic Green's functions of electromagnetic theory. The emphasis is on infinite-medium (or free-space) time-harmonic solutions throughout, thus putting the focus on the physical medium in which the electromagnetic process takes place. The medium's properties enter Maxwell's equations through the constitutive relations, and a comprehensive listing of dyadic Green's functions for which closed-form solutions exist, is given. Presently, the list of media contains (achiral) isotropic, biisotropic (including chiral), generally uniaxial, electrically (or magnetically) gyrotropic, diffusive, and moving media as well as certain plasmas. A critical evaluation of the achievements, successes, limits, and failures of the analytic techniques is provided, and a prognosis is put forward about the future place of analytic methods within the general context of the search for solutions to electromagnetic field problems.