  General Relativity (GR) is consistent with a wide range of
experiments/observations from millimeter scales up to galactic scales and
beyond. However, there are reasons to believe that GR may need to be modified
because it includes singularities (it is an incomplete theory) and also it
requires fine-tuning to explain the accelerating expansion of the universe
through the cosmological constant. Thus, it is important to check various
experiments and observations beyond the above range of scales for possible
hints of deviations from the predictions of GR. If such hints are found it is
important to understand which classes of modified gravity theories are
consistent with them. The goal of this review is to summarize recent progress
on these issues. On sub millimeter scales we show an analysis of the data of
the Washington experiment (Kapner et al. (2007)) searching for modifications of
Newton's Law on sub-millimeter scales and demonstrate that a spatially
oscillating signal is hidden in this dataset. We show that even though this
signal cannot be explained in the context of standard modified theories (viable
scalar tensor and $f(R)$ theories), it is a rather generic prediction of
nonlocal gravity theories. On cosmological scales we review recent analyses of
Redshift Space Distortion data which measure the growth rate of cosmological
perturbations at various redshifts and show that these data are in some tension
with the $\Lambda$CDM parameter values indicated by Planck/2015 CMB data at
about 3$\sigma$ level. This tension can be reduced by allowing for an evolution
of the effective Newton constant that determines the growth rate of
cosmological perturbations. We conclude that even though this tension between
the data and the predictions of GR could be due to systematic/statistical
uncertainties of the data, it could also constitute early hints of a new
gravitational theory.
