Knowledge and language understanding of models evaluated through question answering (QA) has been usually studied on static snapshots of knowledge, like Wikipedia. However, our world is dynamic, evolves over time, and our models' knowledge becomes outdated. To study how semi-parametric QA models and their underlying parametric language models (LMs) adapt to evolving knowledge, we construct a new large-scale dataset, StreamingQA, with human written and generated questions asked on a given date, to be answered from 14 years of time-stamped news articles. We evaluate our models quarterly as they read new articles not seen in pre-training. We show that parametric models can be updated without full retraining, while avoiding catastrophic forgetting. For semi-parametric models, adding new articles into the search space allows for rapid adaptation, however, models with an outdated underlying LM under-perform those with a retrained LM. For questions about higher-frequency named entities, parametric updates are particularly beneficial. In our dynamic world, the StreamingQA dataset enables a more realistic evaluation of QA models, and our experiments highlight several promising directions for future research.