Conformal inference is a powerful tool for quantifying the uncertainty around predictions made by black-box models (e.g. neural nets, random forests). Formally, this methodology guarantees that if the training and test data are exchangeable (e.g. i.i.d.) then we can construct a prediction set $C$ for the target $Y$ such that $P(Y \in C) = 1-\alpha$ for any target level $\alpha$. In this article, we extend this methodology to an online prediction setting where the distribution generating the data is allowed to vary over time. To account for the non-exchangeability, we develop a protective layer that lies on top of conformal inference and gradually re-calibrates its predictions to adapt to the observed changes in the environment. Our methods are highly flexible and can be used in combination with any predictive algorithm that produces estimates of the target or its conditional distribution and without any assumptions on the size or type of the distribution shift. We test our techniques on two real-world datasets aimed at predicting stock market volatility and COVID-19 case counts and find that they are robust and adaptive to real-world distribution shifts.