There is a fascinating open problem in Riemannian Geometry which I would like to advertise here because I do not think that it is as well-known as it deserves to be. Euclid's famous fifth postulate, or more precisely Playfair's version of it, states that, in the Euclidean plane, through every point outside a line $\ell$ there passes one and only one line which does not intersect $\ell$.
Question: Is the Euclidean plane the only complete Riemannian manifold homeomorphic to $R^2$ which satisfies the fifth postulate, i.e., through any point outside a complete geodesic $\gamma$ there passes one and only one complete geodesic which does not intersect $\gamma$?
In other words, does the fifth postulate force the curvature to be zero? The reason this is interesting, or historically significant, is that it was the attempts to reduce the fifth postulate to the other axioms of Euclid, throughout the middle ages, which eventually led to the discovery of non Euclidean geometries and the notion of curvature by Gauss and Riemann.
This problem appears to be originally due to Burns and Knieper in 1991: see the survey paper by Burns and Matveev, which also includes other nice problems on geometry of geodesics. The problem is also mentioned in papers of Croke, and Bangert and Emmerich, and has been studied most recently by Ge, Guijarro, and Solorzano.