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Addressing a wide range of topics, Noncommutative Geometry and Cayley-Smooth Orders explains the theory of Cayley-smooth order in central simple algebras. It provides all of the necessary background material required to understand this theory, including the invariant theoretic description of orders and their centers. The book introduces etale topology and its use in noncommutative algebra, in particular to the study of Azymaya and the local description of algebras via Luna slices. It also offers a discussion on the representations of quivers, including their indecomposable roots, the determination of dimension vectors of simple representations, and general representations.