Is it possible to detect gravitational lensing behind the Moon? Eddington waited for an eclipse to happen to be able to observe gravitational lensing of the stars behind the Sun. And nowadays, amateurs can do the same thing. 

Of course, the Moon is much lighter than the Sun, so it doesn't bend light as much. But since Eddington's time, our observation capabilities have greatly increased. Is it possible to detect the gravitational lensing of stars that are behind the Moon?
The angular deflection caused by the lensing of a distant background object by the Moon is given by
$$\theta \simeq 4 \frac{GM}{Rc^2},$$
where $M$ is the mass of the lensing object and $R$ is the closest projected distance of the ray from the centre of the mass.

For the Moon, the maximum deflection would occur when the ray just grazes the limb of the Moon and would equal 26 micro-arcseconds.

At present the most accurate instruments for measuring precision positions are VLBI radio observations of point sources, where 10 micro-arcsecond relative positional precision can be "routinely achieved" (Reid &amp; Honma 2014). To use this technique you need bright, radio point sources to be close in position to the Moon. This is certainly possible and has been done many times using the Sun as the lensing object, although the claimed accuracies of a short time-series of measurements on single sources is an order of magnitude lower and so probably wouldn't work for the Moon (e.g. Titov et al. 2018 ). 

The Gaia astrometry satellite has likely end-of-mission positional precisions of about 5 micros-arcseconds for bright stars, but it is unclear what the precision is for a single scan (which would be needed because the Moon moves!). However, to counterbalance that, one could also average the lensing effect over many stars surrounding the Moon and of course average this for different stars with the Moon observed at different times over the entire Gaia mission.

Unfortutunately, although in principle this could be done, because of the way that Gaia scans the sky, it is always pointing away from the Moon and so there will be no observations towards the Moon that would be capable of revealing these small deflections (e.g. see here).

Conclusion: Measuring the gravitational deflection of light by the Moon is just out of reach of current observational techniques.