What are the anomalies with General Relativity? General relativity is the current model of gravity which has not yet been disproved. Are there still any anomalies such as the problem of Mercury's orbit during Newtonian gravity period that it failed to explain? If so are there other types of physics to be discovered?
 A: People usually focus on so-called strong field tests of GR, but I'm going to go completely the other direction to extremely weak fields, $a=GM/r^2\lesssim 10^{-10}\,{\rm m}\,{\rm s}^{-2}$.
One example that's been around for a few decades is the rotation curves of galaxies. Whereas GR (the fields are weak, so really it's just the Newtonian limit) would predict from the visible matter distribution that the rotation velocity rises steeply then falls off with radius, most galaxies seem to rise then stay approximately flat. One interpretation of this is that there is a lot of unseen 'dark matter', but an alternative is that in the weak field limit gravity isn't well described by GR. One point in favour of a modification to GR is the 'mass discrepancy-acceleration relation'. This relates the deficit of mass required to explain the observed rotation (i.e. the discrepancy w.r.t. the GR prediction) with the observed circular acceleration. These two quantities correlate very tightly, which could be an argument that it's something to do with the force law rather than the matter distribution. But others argue that the same relation can be explained within the dark matter paradigm (disclaimer: I'm an author of that paper). So it's currently an open problem. I think the only thing that nearly everyone agrees on is that there is some sort of new physics waiting to be explained here.
These issues, and some other weak-field anomalies, are discussed at a fairly accessible level in this recent paper. The author does a good job of pointing out the successes and failures of both the dark matter and modified gravity interpretations.
A: Perhaps the most important one is singularities: GR predicts its own downfall. The reason for this: General relativity is incompatible with quantum mechanics, as it doesn't follow the uncertainty principle. When the coordinate systems are very unstable, no reasonably sensible tensor analysis can be done, hence GR's very foundation is challenged. And this is exactly what happens in singularities, where quantum effects can't be ignored. And already, this approach to quantum gravity has shattered many "classical" assumptions; for example, Hawking radiation predicts emission of photons from a black hole, which classically didn't emit anything.
