One of my lectures, which I had during the Week of Doctoral Students in Prague (in 2011)

We could study General Relativity, when we consider d spatial dimensions (instead of 3). Then the computations are much more involved and there are many open issues in perturbation theory. Of course, such an extended theory is not realized in Nature, because we don’t observe extended extra dimensions. But people are still trying to search for small curved extra dimensions. Here is the presentation (pictures are taken from web).