Building trustless cross-blockchain trading protocols is challenging. Centralized exchanges thus remain the preferred route to execute transfers across blockchains. However, these services require trust and therefore undermine the very nature of the blockchains on which they operate. To overcome this, several decentralized exchanges have recently emerged which offer support for atomic cross-chain swaps (ACCS). ACCS enable the trustless exchange of cryptocurrencies across blockchains, and are the only known mechanism to do so. However, ACCS suffer significant limitations; they are slow, inefficient and costly, meaning that they are rarely used in practice. We present XCLAIM: the first generic framework for achieving trustless and efficient cross-chain exchanges using cryptocurrency-backed assets (CbAs). XCLAIM offers protocols for issuing, transferring, swapping and redeeming CbAs securely in a non-interactive manner on existing blockchains. We instantiate XCLAIM between Bitcoin and Ethereum and evaluate our implementation; it costs less than USD 0.50 to issue an arbitrary amount of Bitcoin-backed tokens on Ethereum. We show XCLAIM is not only faster, but also significantly cheaper than atomic cross-chain swaps. Finally, XCLAIM is compatible with the majority of existing blockchains without modification, and enables several novel cryptocurrency applications, such as cross-chain payment channels and efficient multi-party swaps.