This chapter illustrates the conditions which are necessary in order to obtain one solution to Maxwell’s equations. In the complex-vector representation of time-harmonic vectors, the uniqueness theorem has to be modified, slightly but in a non-trivial manner. The familiarity with reciprocity may induce the reader to believe that all the consequences of the theorem match easily with intuition. The chapter provides a corollary of the equivalence theorem. The proof of the equivalence theorem required the new concept of magnetic currents, which is a useful mathematical tool, with no experimental significance. Geometrical symmetries of the region where an electromagnetic field is defined, if properly exploited, can make its determination significantly simpler. The TE–TM decomposition of an e.m. field is the strategic starting point for waveguide theory, both at microwave and optical frequencies.