GR is a “perfect” theory that in the last century successfully undergo
many stringent observational tests.
Its unchallenged position in both small (Solar System) scales and
strong field limits (i.e. BH and NS) made it a natural
candidate for the fundamental building block for standard cosmological
model (LCDM).
However, GR has not been so far rigorously tested in the very
low-densities and at intergalactic scales.
Thus the reality doors are still quite open for many prospective
alternative theories commonly dubbed Modified Gravity.
I shall recapture the general landscape of the currently very vibrant
field of modified gravity theories.
I will also discuss why is it absolutely crucial to find and provide
as many as possible independent cosmological tests
of GR and its competitors.
Finally, I will present some of quite non-standard cosmological tests
of gravity that me and my collaborators have put forward recently.
These include hierarchical amplitudes (scaled higher-order central
moments of the density field) and halo/galaxy clustering ratios.