  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.
