Theoretical Methods for Giant Resonances

The Random Phase Approximation (RPA) and its variations and extensions are, without any doubt, the most widely used tools to describe Giant Resonances within a microscopic theory. In this chapter, we will start by discussing how RPA comes out naturally if one seeks a state with a harmonic time dependence in the space of one particle-one hole excitations on top of the ground state. It will be also shown that RPA is the simplest approach in which a ``collective'' state emerges. These are basic arguments that appear in other textbooks but are also unavoidable as a starting point for further discussions. In the rest of the chapter, we will give emphasis to developments that have taken place in the last decades: alternatives to RPA like the Finite Amplitude Method (FAM), state-of-the-art calculations with well-established Energy Density Functionals (EDFs), and progress in {\em ab initio} calculations. We will discuss extensions of RPA using as a red thread the various enlargements of the one particle-one hole model space. The importance of the continuum, and the exclusive observables like the decay products of Giant Resonances, will be also touched upon.