Performance limits for electromagnetic devices can be determined from sum rules and optimization over equivalent sources. Sum rules are derived by identifying a passive system with a Herglotz function and using integral identities to relate the dynamic response with its low and high-frequency asymptotic expansions. These bounds and identities are of great interest in many areas of physics and engineering partly because of their simplicity and close form expressions. Optimization over equivalent sources complements the sum rule limits by providing single frequency bounds and a flexibility to incorporate different types of information as constraints in the optimization problems. Here, an overview of Herglotz function and optimization-based bounds in EM is presented with examples from metamaterials, antennas, and scattering. We also compare the theoretical results with state-of-the-art designs. We also discuss different ways to combine the two approaches.