Fluorescence microscopy is an invaluable tool in biology, yet its performance is compromised when the wavefront of light is distorted due to optical imperfections or the refractile nature of the sample. Such optical aberrations can dramatically lower the information content of images by degrading image contrast, resolution, and signal. Adaptive optics (AO) methods can sense and subsequently cancel the aberrated wavefront, but are too complex, inefficient, slow, or expensive for routine adoption by most labs. Here we introduce a rapid, sensitive, and robust wavefront sensing scheme based on phase diversity, a method successfully deployed in astronomy but underused in microscopy. Our method enables accurate wavefront sensing to less than λ/35 root mean square (RMS) error with few measurements, and AO with no additional hardware besides a corrective element. After validating the method with simulations, we demonstrate calibration of a deformable mirror > 100-fold faster than comparable methods (corresponding to wavefront sensing on the ~100 ms scale), and sensing and subsequent correction of severe aberrations (RMS wavefront distortion exceeding λ/2), restoring diffraction-limited imaging on extended biological samples.