We propose a new inference framework called localized conformal prediction. It generalizes the framework of conformal prediction by offering a single-test-sample adaptive construction that emphasizes a local region around this test sample, and can be combined with different conformal score constructions. The proposed framework enjoys an assumption-free finite sample marginal coverage guarantee, and it also offers additional local coverage guarantees under suitable assumptions. We demonstrate how to change from conformal prediction to localized conformal prediction using several conformal scores, and we illustrate a potential gain via numerical examples.