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This report documents the program and the outcomes of Dagstuhl Seminar 21431 "Rigorous Methods for Smart Contracts". Blockchain technologies have emerged as an exciting field for both researchers and practitioners focusing on formal guarantees for software. It is arguably a "once in a lifetime" opportunity for rigorous methods to be integrated in audit processes for parties deploying smart contracts, whether for fund raising, securities trading, or supply-chain management. Smart contracts are programs managing cryptocurrency accounts on a blockchain. Research in the area of smart contracts includes a fascinating combination of formal methods, programming-language semantics, and cryptography. First, there is vibrant development of verification and program-analysis techniques that check the correctness of smart-contract code. Second, there are emerging designs of programming languages and methodologies for writing smart contracts such that they are more robust by construction or more amenable to analysis and verification. Programming-language abstraction layers expose low-level cryptographic primitives enabling developers to design high-level cryptographic protocols. Automated-reasoning mechanisms present a common underlying enabler; and the specific needs of the smart-contract world offer new challenges. This workshop brought together stakeholders in the aforementioned areas related to advancing reliable smart-contract technologies.