Cell-resolution optical imaging methods, such as confocal microscopy and full-field optical coherence tomography, capture flat optical sections of the sample. If the sample is curved, the optical field sections through several sample layers, and the view of each layer is reduced. Here we present curved-field optical coherence tomography, capable of capturing optical sections of arbitrary curvature. We test the device on a challenging task of imaging the human cornea in vivo and achieve a 10× larger viewing area comparing to the clinical state-of-the-art. This enables more precise cell and nerve counts, opening a path to improved monitoring of corneal and general health conditions (e.g., diabetes). The method is non-contact, compact, and works in a single fast shot (3.5 ms), making it readily available for use in optical research and clinical practice.