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The fast adaptation capability of deep neural networks in non-stationary environments is critical for online time series forecasting. Successful solutions require handling changes to new and recurring patterns. However, training deep neural forecaster on the fly is notoriously challenging because of their limited ability to adapt to non-stationary environments and the catastrophic forgetting of old knowledge. In this work, inspired by the Complementary Learning Systems (CLS) theory, we propose Fast and Slow learning Networks (FSNet), a holistic framework for online time-series forecasting to simultaneously deal with abrupt changing and repeating patterns. Particularly, FSNet improves the slowly-learned backbone by dynamically balancing fast adaptation to recent changes and retrieving similar old knowledge. FSNet achieves this mechanism via an interaction between two complementary components of an adapter to monitor each layer's contribution to the lost, and an associative memory to support remembering, updating, and recalling repeating events. Extensive experiments on real and synthetic datasets validate FSNet's efficacy and robustness to both new and recurring patterns. Our code is available at \url{https://github.com/salesforce/fsnet}.