We introduce GaussianAvatars, a new method to create photorealistic head avatars that are fully controllable in terms of expression, pose, and viewpoint. The core idea of our method is a dynamic 3D representation based on 3D Gaussian splats that are rigged to a parametric morphable face model. This combination facilitates photorealistic rendering while also allowing for precise animation control via the underlying parametric model, e.g., through expression transfer from a driving sequence of a different person or by manually changing the morphable model parameters. In addition to the geometry of the morphable model itself, we optimize for explicit displacement offsets to obtain a more accurate geometric representation. Each splat location is then parameterized by a local coordinate frame to compensate for inaccuracies. During avatar reconstruction, we jointly optimize for the morphable model parameters and Gaussian splat parameters in an end-to-end fashion. We demonstrate the animation capabilities of our photorealistic avatar in several challenging scenarios. For instance, we show reenactments from a driving video, where our method outperforms existing works by a significant margin.