We introduce the theory of open quantum systems and quantum entanglement. We start by studying the states, the dynamics, and the observables of open quantum systems. The central notions of the theory are quantum entanglement and its measure, the von Neumann entropy. The geometry of the open quantum states is described in detail: we discuss Schmidt decomposition, purification of mixed states, the Schrödinger mixing theorem, the Bures metric geometry of mixed states, fidelity, and the Uhlmann theorem. Then we investigate quantum entanglement and EPR states. We discuss the EPR and GHZH gedanken experiments to highlight the non-classical properties of quantum entanglement, and explain how it should be seen as a resource which allows us to perform classically impossible tasks. We review the classical information theory of Shannon and Shannon entropy; we prove the Shannon noiseless and noisy coding theorems. Finally we define the von Neumann quantum entropy and discuss in detail its properties and applications. The quantum relative entropy is also introduced. We conclude with quantum information theory. We prove the quantum noiseless coding theorem.