Abstract: Transactional memory (TM) is an intensively studied synchronisation paradigm with many proposed implementations in software and hardware, and combinations thereof. However, TM under relaxed memory, e.g., C11 (the 2011 C/C++ standard) is still poorly understood, lacking rigorous foundations that support verifiable implementations. This paper addresses this gap by developing TMS2-RA, a relaxed operational TM specification. We integrate TMS2-RA with RC11 (the repaired C11 memory model that disallows load-buffering) to provide a formal semantics for TM libraries and their clients. We develop a logic, TARO, for verifying client programs that use TMS2-RA for synchronisation. We also show how TMS2-RA can be implemented by a C11 library, TML-RA, that uses relaxed and release-acquire atomics, yet guarantees the synchronisation properties required by TMS2-RA. We benchmark TML-RA and show that it outperforms its sequentially consistent counterpart in the STAMP benchmarks. Finally, we use a simulation-based verification technique to prove correctness of TML-RA. Our entire development is supported by the Isabelle/HOL proof assistant.