We present a novel approach to probabilistic time series forecasting that combines state space models with deep learning. By parametrizing a per-time-series linear state space model with a jointly-learned recurrent neural network, our method retains desired properties of state space models such as data efficiency and interpretability, while making use of the ability to learn complex patterns from raw data offered by deep learning approaches. Our method scales gracefully from regimes where little training data is available to regimes where data from millions of time series can be leveraged to learn accurate models. We provide qualitative as well as quantitative results with the proposed method, showing that it compares favorably to the state-of-the-art.