Lectures on Matrix Field Theory These lecture notes provide a systematic introduction to matrix models of quantum field theories with non commutative and fuzzy geometries. The book initially focuses on the matrix formulation of non commutative and fuzzy spaces followed by a description of the non perturbative treatment of the corresponding field theories. As an example the phase structure of non commutative phi four theory is treated in great detail with a separate chapter on the multitrace approach. The last chapter offers a general introduction to non commutative gauge theories while two appendices round out the text. Primarily written as a self study guide for postgraduate students with the aim of pedagogically introducing them to key analytical and numerical tools as well as useful physical models in applications these lecture notes will also benefit experienced researchers by providing a reference guide to the fundamentals of non commutative field theory with an emphasis on matrix models and fuzzy geometries Preface Introductory Remarks The Non Commutative Moyal Weyl Spaces Rd The Fuzzy Sphere Quantum Non Commutative Phi Four The Multitrace Approach Non Commutative Gauge Theory Appendix A The Landau States Appendix B The Traces TrtAtB and TrtAtBtCtD Index