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In the photonic design problem, a scientist or engineer chooses the physical parameters of a device to best match some desired device behavior. Many instances of the photonic design problem can be naturally stated as a mathematical optimization problem that is computationally difficult to solve globally. Because of this, several heuristic methods have been developed to approximately solve such problems. These methods often produce very good designs, and, in many practical applications, easily outperform 'traditional' designs that rely on human intuition. Yet, because these heuristic methods do not guarantee that the approximate solution found is globally optimal, the question remains of just how much better a designer might hope to do. This question is addressed by performance bounds or impossibility results, which determine a performance level that no design can achieve. We focus on algorithmic performance bounds, which involve substantial computation to determine. We illustrate a variety of both heuristic methods and performance bounds on two examples. In these examples (and many others not reported here) the performance bounds show that the heuristic designs are nearly optimal, and can considered globally optimal in practice. This review serves to clearly set up the photonic design problem and unify existing approaches for calculating performance bounds, while also providing some natural generalizations and properties.