Since the work of Isaac Newton, the development of quantum theory at the beginning of the 20th century was the greatest advance in physical science. Arguably, quantum mechanics has now become the most successful theory of all time. This course for upper level students of physics and chemistry presents a brief history of quantum theory, followed by an introduction to the standard formalism of quantum mechanics, e.g. the postulates of quantum theory, the properties of linear operators, Dirac notation etc. The gained knowledge is then applied to the physics of atoms and molecules and a range of spectroscopic methods, including electron spin resonance, nuclear magnetic resonance and rotational, vibrational and electronic spectroscopy.