Reliable object grasping is a crucial capability for autonomous robots. However, many existing grasping approaches focus on general clutter removal without explicitly modeling objects and thus only relying on the visible local geometry. We introduce CenterGrasp, a novel framework that combines object awareness and holistic grasping. CenterGrasp learns a general object prior by encoding shapes and valid grasps in a continuous latent space. It consists of an RGB-D image encoder that leverages recent advances to detect objects and infer their pose and latent code, and a decoder to predict shape and grasps for each object in the scene. We perform extensive experiments on simulated as well as real-world cluttered scenes and demonstrate strong scene reconstruction and 6-DoF grasppose estimation performance. Compared to the state of the art, CenterGrasp achieves an improvement of 38.5 mm in shape reconstruction and 33 percentage points on average in grasp success. We make the code and trained models publicly available at http://centergrasp.cs.uni-freiburg.de.