Our experimental results demonstrate that full-field hard-X-ray microscopy is finally able to investigate the internal structure of cells in tissues. This result was made possible by three main factors: the use of a coherent (synchrotron) source of X-rays, the exploitation of contrast mechanisms based on the real part of the refractive index and the magnification provided by high-resolution Fresnel zone-plate objectives. We specifically obtained high-quality microradiographs of human and mouse cells with 29nm Rayleigh spatial resolution and verified that tomographic reconstruction could be implemented with a final resolution level suitable for subcellular features. We also demonstrated that a phase retrieval method based on a wave propagation algorithm could yield good subcellular images starting from a series of defocused microradiographs. The concluding discussion compares cellular and subcellular hard-X-ray microradiology with other techniques and evaluates its potential impact on biomedical research.