Most texts on quantum mechanics are primarily designed for physicists. This one, taking a somewhat different approach, is geared toward upper-level undergraduates and graduate students in applied mathematics, and offers a practical and accessible overview of the most powerful and widely used tool of modern mathematical physics. The author develops the subject in a systematic and logical manner from a minimal set of axioms. These axioms are either suggested by experience of the physical world or necessitated by the mathematical model's requirement of internal consistency -- an approach that will appeal to both applied mathematicians and mathematical physicists.The principles of the subject receive the primary focus throughout, and special physical problems appear strictly for the purpose of illustrating these principles. Suggestions for the solution of a large number of such problems are contained in the sets of exercises, of which there are more than a hundred. The theory is first developed for observables, such as the spin, which have only a finite number of eigenvalues. Later, the theory is extended to observables having continuous spectra; by this means, the theory's chief features are exhibited in the simplest possible mathematical context.