A general algebraic formalism for the mathematical modeling of physical systems is sketched. This formalism is sufficiently general to encompass classical and quantum-mechanical models. It is then explained in which way quantum theory differs in an essential way from classical theory and what it is that quantum theory tells us about Nature when suitable experiments are made. Some of the seemingly confusing aspects of quantum theory are highlighted, and it is explained why they actually should not confuse us. A variety of concrete problems of potential interest to mathematicians will be mentioned.