We present ZeroEGGS, a neural network framework for speech-driven gesture generation with zero-shot style control by example. This means style can be controlled via only a short example motion clip, even for motion styles unseen during training. Our model uses a Variational framework to learn a style embedding, making it easy to modify style through latent space manipulation or blending and scaling of style embeddings. The probabilistic nature of our framework further enables the generation of a variety of outputs given the same input, addressing the stochastic nature of gesture motion. In a series of experiments, we first demonstrate the flexibility and generalizability of our model to new speakers and styles. In a user study, we then show that our model outperforms previous state-of-the-art techniques in naturalness of motion, appropriateness for speech, and style portrayal. Finally, we release a high-quality dataset of full-body gesture motion including fingers, with speech, spanning across 19 different styles.