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Recent work on 3D-aware image synthesis has achieved compelling results using advances in neural rendering. However, 3D-aware synthesis of face dynamics hasn't received much attention. Here, we study how to explicitly control generative model synthesis of face dynamics exhibiting non-rigid motion (e.g., facial expression change), while simultaneously ensuring 3D-awareness. For this we propose a Controllable Radiance Field (CoRF): 1) Motion control is achieved by embedding motion features within the layered latent motion space of a style-based generator; 2) To ensure consistency of background, motion features and subject-specific attributes such as lighting, texture, shapes, albedo, and identity, a face parsing net, a head regressor and an identity encoder are incorporated. On head image/video data we show that CoRFs are 3D-aware while enabling editing of identity, viewing directions, and motion.