These graduate lectures aim to give a relatively detailed introduction to different facets of modified gravity models.
Initially, in the first part, we will motivate why, with the discovery of the late time acceleration of the universe, there has recently been a huge amount of activity devoted to developing models which modify gravity on cosmological scales, and hence lead us to revise the standard model of cosmology based on general relativity.
In the second part we will discuss some of the generic features of modified gravity models: the existence of extra degrees of freedom, their stability (and questions related to the Ostrogradski ghost), and how these degrees of freedom must be screened on smaller scales in order to evade the tight gravitational tests in the solar system and the laboratory.
The third part will present in detail, some specific models of modified gravity. In particular we will consider massive gravity, as well as f(R), and the so-called Horndeski and beyond Horndeski theories. All of these have attracted a great deal of interest recently.
Finally, in the last part, we discuss how modifications of gravity can be tested on cosmological scales, focusing on linear and non-linear cosmological perturbations. The latest constraints on deviations from the standard Lambda-CDM will also be reviewed.