The Coq proof assistant provides a formal language to write mathematical definitions, executable algorithms, and theorems, together with an environment for semi-interactive development of machine-checked proofs. Typical applications include the certification of properties of programming languages (e.g., the CompCert compiler certification project and the Bedrock verified low-level programming library), the formalization of mathematics (e.g., the full formalization of the Feit-Thompson theorem and homotopy type theory) and teaching.