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This book provides an introduction the propagator theory. Propagators are two-parameter families of linear operators, also known as evolution operators, evolutions families, non-autonomous sernigroups, etc., which are often used as mathematical models of a system evolving in a changing environment. Although this book concerns such diverse subjects as analysis, semigroup theory, probability theory, mathematical physics, and partial differential equations, it is unified by the Feynman-Kac propagator which describes the evolution of a physical system in the presence of time-dependent absorption and excitation. The theory of Feynman-Kac propagators with potentials from non-autonomous Kato classes is presented in the book for the first time. This book is suitable as an advanced textbook for graduate courses.