In this thesis we will give a presentation of a graphical/diagrammatic calculus for quantum systems involving interacting quantum observables such as multi-partite systems of qubits, the ZX-Calculus. Unlike the Hilbert space formulation of quantum mechanics, the ZX-Calculus is based on category theory, more specically on the notion of a compact dagger symmetric monoidal category and as a consequence the graphical language associated with such a category is inherited by the calculus. This enables us to think about and deal with many calculations in quantum computation and information in a purely graphical and intuitive fashion. Although being formulated in a more general mathematical framework, huge parts of the Hilbert space formulation of quantum mechanics can be extracted from the ZX-calculus. In this thesis we will begin by giving a motivation for the need of such a calculus and then key concepts of category theory will be introduced in an intuitive manner in order to understand the ZX-calculus that will be presented afterwards. We then apply the calculus to'model' and describe certain quantum circuits and quantum teleportation.