The high photon flux and femtosecond pulse duration of hard X-ray free-electron lasers have spurred a large variety of novel and fascinating experiments in physical, chemical and biological sciences. Many of these experiments depend fundamentally on a clean, well-defined wavefront. Here we explore the wavefront properties of hard X-ray free-electron laser radiation by means of a grating interferometer, from which we obtain shot-to-shot wavefront information with an excellent angular sensitivity on the order of ten nanoradian. The wavefront distortions introduced by optical elements are observed in-situ and under operational conditions. The source-point position and fluctuations are measured with unprecedented accuracy in longitudinal and lateral direction, both during nominal operation and as the X-ray free-electron laser is driven into saturation.