Quantum Field Theory (QFT) is one of the pillars of modern scientific knowledge, and it is applied in many different areas of physics, from the theory of elementary particles to condensed matter. However, most QFTs cannot be solved exactly and one has to perform some type of approximation in order to extract information from them. The standard approach is a perturbative expansion in a small coupling constant, implemented diagrammatically through Feynman diagrams. This approach has been enormously successful, but it is also insufficient to address many important phenomena which are supposed to be described by QFT.

In this book I give an introduction to two methods in QFT which go beyond the standard perturbative framework: instantons and the large Nexpansion. Both are quite general, and they have led to many useful insights on the non-perturbative aspects of QFT. The two questions that I will address in this book are the following. What are the kinds of phenomena in a QFT in which we are fundamentally misled if we use conventional perturbation theory? What kind of intuition can we get on these effects by using instantons or large Nphysics?

It should be said from the very beginning that the main intention of this text is to give a pedagogicalintroduction to these topics. I would like to provide a useful toolbox intended for graduate students and beginners. In line with this, I provide many computational details which are usually skipped in the original literature and in many textbooks.

The treatment of the subject pays tribute to my professional bias: I am a mathematical physicist, not a phenomenologist. Therefore, in this book, after introducing an idea or a technique, I try to illustrate it with a model or example where this idea is implemented in a nice mathematical way, independently of its relevance to measurable physics. This involves typically looking at models in a low number of dimensions (either two, one or even zero). However, I hope that some of the material in this book will be useful for people with a more phenomenological orientation. After all, toy, solvable models should be useful to everybody, and they typically provide useful insights for more realistic applications.