In this paper, we consider the problem of cross-chain transactions where parties that do not trust each other safely exchange digital assets across blockchains. Open blockchains models are decentralized ledgers that keep records of transactions. They are comparable with distributed account books. While they have proven their potential as a store of value, exchanging assets across several blockchains remains a challenge. Our paper proposes a new protocol, R-SWAP, for cross-chain swaps that outperforms existing solutions. Our protocol is built on top of two abstractions: relays and adapters that we formalize for the first time in this paper. Furthermore, we prove the correctness of R-SWAP and analytically evaluate its performances, in terms of cost and latency. Moreover, we evaluate the performances of R-SWAP in two case studies showing the generality of our approach: atomic swaps between Ethereum and Bitcoin (two popular permissionless blockchains) and atomic swaps between Ethereum and Tendermint (one permissionless and one permissioned blockchain).