Cryogenic electron tomography is a technique for imaging biological samples in 3D. A microscope collects a series of 2D projections of the sample, and the goal is to reconstruct the 3D density of the sample called the tomogram. Reconstruction is difficult as the 2D projections are noisy and can not be recorded from all directions, resulting in a missing wedge of information. Tomograms conventionally reconstructed with filtered back-projection suffer from noise and strong artifacts due to the missing wedge. Here, we propose a deep-learning approach for simultaneous denoising and missing wedge reconstruction called DeepDeWedge. The algorithm requires no ground truth data and is based on fitting a neural network to the 2D projections using a self-supervised loss. DeepDeWedge is simpler than current state-of-the-art approaches for denoising and missing wedge reconstruction, performs competitively and produces more denoised tomograms with higher overall contrast.