Several mathematicians have maintained that mathematical beauty plays an important role in mathematical research. This raises the problem: What is mathematical beauty? This chapter supports the view that a piece of mathematics, demonstration or theorem, is beautiful when it provides understanding, meaning by this the recognition of the fitness of the parts to each other and to the whole. Specifically, a demonstration is beautiful when it is clear what the whole idea of the demonstration is, what the contribution of each part of the demonstration to the whole idea is, and why such contribution is essential. A theorem is beautiful when it is clear what the content of the theorem is, what the contribution of each concept to the content is, and what relations the theorem establishes between such concepts. Mathematical beauty thus meant can have a role in mathematical discovery, because it can guide the mathematician in selecting which hypothesis to consider and which to disregard.