This chapter provides a review of the mathematical theory of linear quantum systems, which is based on the Hudson–Parthasarathy quantum stochastic calculus as a mathematical tool for describing Markov open quantum systems interacting with external propagating quantum fields. A precise definition of linear quantum systems is given as well as quantum stochastic differential equations representing their linear equation of motion in the Heisenberg picture. The important notion of physical realizability for linear quantum stochastic differential equations is introduced, and necessary and sufficient conditions for physical realizability reviewed. Complete parameterizations for linear quantum systems are given, and transfer functions defined. Also, the special class of completely passive linear quantum systems is introduced and the notion of stability for linear quantum systems is developed.