Instabilities in field theories: Lecture notes with a view into modified gravity

Modified theories of gravity usually present new degrees of freedom, as well as higher order derivatives, wrong signs in certain terms and complicated couplings already present in the Lagrangian from the beginning or originated by the field redefinitions needed to reach an Einstein frame. As a consequence, they are very prone to present dynamical instabilities that could spoil any attempt to construct viable models within these frameworks. In these three lectures we introduce the most common types of instabilities that appear in field theory as well as some techniques to detect them, and supplement these contents with several examples. The goal is to understand the implications of having such pathological behaviors and the application of these notions to modified theories of gravity.