Many precise tests are available for general relativity (GR) in the "weak field" regime, but only observations of the emission from very close to the horizon event of a black hole can probe GR in "strong field" conditions, where the distorsions of space-time due to gravity play a major role. The hard X-ray emission of active galactic nuclei is produced in this inner region, and is therefore an optimal observational tool to probe GR. However, these tests have been proven to be inconclusive, because of a strong degeneracy in the X-ray spectra of AGN between the intrinsic emission and the complex absorption/reflection components due to the circumnuclear medium. I will review the main diagnostic methods developed in the past few years to investigate this issue, focusing on time-resolved spectral analysis of variable sources, and I will discuss the impact in this field of the recently launched hard X-ray telescope NuSTAR. I will show how the availability of high quality broad-band X-ray spectra, and a time-resolved spectral analysis can remove the main systematic uncertainties in GR tests, provide precise measurements of black hole spins, and reveal the structure of the circumnuclear X-ray absorber.