The accelerated expansion of the Universe is believed to have occurred at least twice - once in the early (inflationary) and the present epoch.
The underlying mechanism behind the cosmic acceleration remains unknown.
Motivated by this mystery, significant theoretical progress has been made in (classical) gravitational physics, including the discovery of classical mechanisms of cosmic acceleration within gravity theories beyond the conventional general relativity.
Remarkably, some scalar-tensor theories, a specific extension with one scalar d.o.f. to the general relativity, exhibit nonlinear screening mechanism on stellar scales whereas not on cosmological scales.
This enables scalar-tensor theories to mitigate the stringent constraints from the solar system.
Hence testing gravity theories on cosmological scales has drawn attention.
In the mean time, high-precision measurements of the cosmic microwave background, the three-dimensional distribution of galaxies, gravitational lensing, and gravitational waves etc. have enabled concrete tests of the concordance $\Lambda$ CDM model, as well as gravitational physics on cosmological scales.
Foreseeing observational programmes of the late 2020s to 2030s, we will review what people aim to understand about gravity and the nature of cosmic acceleration.
If the time is permitted, we will show our latest works.