We introduce VIVE3D, a novel approach that extends the capabilities of image-based 3D GANs to video editing and is able to represent the input video in an identity-preserving and temporally consistent way. We propose two new building blocks. First, we introduce a novel GAN inversion technique specifically tailored to 3D GANs by jointly embedding multiple frames and optimizing for the camera parameters. Second, besides traditional semantic face edits (e.g. for age and expression), we are the first to demonstrate edits that show novel views of the head enabled by the inherent properties of 3D GANs and our optical flow-guided compositing technique to combine the head with the background video. Our experiments demonstrate that VIVE3D generates high-fidelity face edits at consistent quality from a range of camera viewpoints which are composited with the original video in a temporally and spatially-consistent manner.