This research note contains some recent results in the theory of strongly continuous semigroups of linear operators. Topics covered include the Feynman-Kac formalism with applications to Schrödinger operators, positivity preserving semigroups, quadratic form theory, holomorphic semigroups and bounded one-parameter groups. An appendix deals with some aspects of Feynman path integrals. A number of previously inaccessible results in semigroup theory are presented with full proofs. This work will be useful both for those undertaking research as well as for graduate students; a basic knowledge of functional analysis and probability theory are the only prerequisites.

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