Two things that have really fascinated me over the past few years are the Pioneer anomaly and dark energy/matter. Apparently the pioneer spacecraft are not traveling exactly on the trajectories expected. This might not mean anything or it could be an indication that our understanding of gravity has to be modified. You can read about the Pioneer anomaly on Wikipedia:
A listing of papers on the archive that discuss the pioneer anomaly can be found here:
Of course so-called dark energy has revived the need for Einstein's cosmological constant term. Or does it? Alternative theories have been proposed which modify gravity on large scales. If you're more technically inclined you might want to check out a recent paper titled Distinguishing Modified Gravity from Dark Energy. The abstract reads:
The acceleration of the universe can be explained either through dark energy or through the modification of gravity on large scales. In this paper we investigate modified gravity models and compare their observable predictions with dark energy models. Modifications of general relativity are expected to be scale-independent on super-horizon scales and scale-dependent on sub-horizon scales. For scale-independent modifications, utilizing the conservation of the curvature scalar and a parameterized post-Newtonian formulation of cosmological perturbations, we derive results for large scale structure growth, weak gravitational lensing, and cosmic microwave background anisotropy. For scale-dependent modifications, inspired by recent $f(R)$ theories we introduce a parameterization for the gravitational coupling $G$ and the post-Newtonian parameter $\gamma$. These parameterizations provide a convenient formalism for testing general relativity. However, we find that if dark energy is generalized to include both entropy and shear stress perturbations, and the dynamics of dark energy is unknown a priori, then modified gravity cannot in general be distinguished from dark energy using cosmological linear perturbations.
It is available as a PDF at this link:
Here is a paper recently posted on the archive which reviews tests of general relativity in the solar system:
The abstract reads:
Tests of gravity performed in the solar system show a good agreement with general relativity. The latter is however challenged by observations at larger, galactic and cosmic, scales which are presently cured by introducing ``dark matter'' or ``dark energy''. A few measurements in the solar system, particularly the so-called ``Pioneer anomaly'', might also be pointing at a modification of gravity law at ranges of the order of the size of the solar system. The present lecture notes discuss the current status of tests of general relativity in the solar system. They describe metric extensions of general relativity which have the capability to preserve compatibility with existing gravity tests while opening free space for new phenomena. They present arguments for new mission designs and new space technologies as well as for having a new look on data of existing or future experiments.