Quantum mechanics is widely recognized as the basic law which governs all of nature, including all materials and devices. It has always been essential to the understanding of material properties, and as devices become smaller it is also essential for studying their behaviour. Nevertheless, only a small fraction of graduate engineers and materials scientists take a course giving a systematic presentation of the subject. The courses for physics students tend to focus on the fundamentals and formal background, rather than on application, and do not fill the need. This text has been designed to fill the gap. The book covers those parts of quantum theory which may be necessary for a modern engineer. It focuses on the approximations and concepts which allow estimates of the entire range of properties of nuclei, atoms, molecules and solids, as well as the behaviour of lasers and other quantum-optic devices. It may be useful to graduate students in physics, whose courses on quantum theory tend not to include any of these applications. The material has been the basis of a course taught to graduate engineering students at Stanford University.