In this paper we investigate the advantages of using the product-integral calculus rather than the usual successive approximation perturbative expansion in three branches of quantum physics. The mathematics of the calculus is fully developed, and its superiority over the perturbative expansion is explored in specific examples from physics. All of time-dependent perturbation theory in quantum mechanics can be written in terms of the calculus, as well as the most general solution for the density operator in quantum statistical mechanics.