Conformal prediction is a popular, modern technique for providing valid predictive inference for arbitrary machine learning models. Its validity relies on the assumptions of exchangeability of the data, and symmetry of the given model fitting algorithm as a function of the data. However, exchangeability is often violated when predictive models are deployed in practice. For example, if the data distribution drifts over time, then the data points are no longer exchangeable; moreover, in such settings, we might want to use an algorithm that treats recent observations as more relevant, which would violate the assumption that data points are treated symmetrically. This paper proposes new methodology to deal with both aspects: we use weighted quantiles to introduce robustness against distribution drift, and design a new technique to allow for algorithms that do not treat data points symmetrically, with theoretical results verifying coverage guarantees that are robust to violations of exchangeability. This work is joint with Emmanuel Candes, Aaditya Ramdas, and Ryan Tibshirani.