High-contrast imaging instruments need extreme wavefront control to directly image exoplanets. This requires highly sensitive wavefront sensors which optimally make use of the available photons to sense the wavefront. Here, we propose to numerically optimize Fourier-filtering wavefront sensors using automatic differentiation. First, we optimize the sensitivity of the wavefront sensor for different apertures and wavefront distributions. We find sensors that are more sensitive than currently used sensors and close to the theoretical limit, under the assumption of monochromatic light. Subsequently, we directly minimize the residual wavefront error by jointly optimizing the sensing and reconstruction. This is done by connecting differentiable models of the wavefront sensor and reconstructor and alternatingly improving them using a gradient-based optimizer. We also allow for nonlinearities in the wavefront reconstruction using Convolutional Neural Networks, which extends the design space of the wavefront sensor. Our results show that optimization can lead to wavefront sensors that have improved performance over currently used wavefront sensors. The proposed approach is flexible, and can in principle be used for any wavefront sensor architecture with free design parameters.