We introduce a microscopic method that determines quantitative optical properties beyond the optical diffraction limit and allows direct imaging of unstained living biological specimens. In established holographic microscopy, complex fields are measured using interferometric detection, allowing diffraction-limited phase measurements. Here, we show that non-invasive optical nanoscopy can achieve a lateral resolution of 90 nm by using a quasi-2 pi-holographic detection scheme and complex deconvolution. We record holograms from different illumination directions on the sample plane and observe subwavelength tomographic variations of the specimen. Nanoscale apertures serve to calibrate the tomographic reconstruction and to characterize the imaging system by means of the coherent transfer function. This gives rise to realistic inverse filtering and guarantees true complex field reconstruction. The observations are shown for nanoscopic porous cell frustule (diatoms), for the direct study of bacteria (Escherichia coil), and for a time-lapse approach to explore the dynamics of living dendritic spines (neurones).