The problem of moments received its first systematic treatment in the late 19th and early 20th centuries through the works of P. L. Chebyshev, A. A. Markov, T. Stieltjes, H. Hamburger, P. Nevannlina, M. Riesz, T. Carleman and many others. In spite of its quite old origin, it continues to exert profound influence on the development of analysis and its applications to a wide variety of fields. In particular, the theory of systems and control is no exception. The aim of this course is to introduce the main concepts and techniques related to the non-harmonic moment problem for the exact and approximate controllability of infinite dimensional systems.