First of all, one could wonder, why should we modify Einstein’s theory, that seems to pass so many experimental test ? In fact, while it does pass many tests, as we pointed here, most of these tests are in the weak field regime. The strong curvature regime is actually not so well tested. The best approaches to dates involve observations of the double pulsar. At the galactic scale already, we don’t understand the observed velocity curves assuming our standard gravity model leading to the assumption of an extra unseen matter (dark matter). Planck‘s result tends to favor the Lambda-CDM model, with CDM standing for Cold Darlk Matter (Lambda is the cosmological constant).
This Dark Matter is a form of matter that interacts essentially gravitationally only. Interestingly, this effect can be understood in terms of modified gravity models. Let us transpose the problem back in time: Mercury’s perihelion advance lead to assuming the existence of an unseen mass (Vulcan), and the effect was finally understood with GR. Here the situation can be similar and there are many attemps to explain dark matter with alternative gravity scenario (see for instance this, among others).
There are many ways to modify Einstein’s theory of gravity and we’ll describe only a few of them in the following subsections