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String theory is the most promising candidate for the theory unifying all interactions including gravity. Since it has an extremely difficult dynamics, many important insights have come from the study of its low-dimensional versions. A particularly interesting case is 2D string theory, which has a very rich structure, but at the same time is solvable. Its complete solution in the simplest linear dilaton background was obtained using its representation as Matrix Quantum Mechanics, which provides a very powerful technique and reveals the integrability hidden in the usual CFT formulation. This book presents a review of these results and extends them to non-trivial backgrounds of 2D string theory. It shows how MQM incorporates the perturbations changing the background, their integrability, and various results on correlation functions, thermodynamic behavior, structure of the target space and non-perturbative effects in the deformed background. The book is recommended for students andresearchers working in theoretical and mathematical physics.