The family of bronzes HxReO3, HxMoO3 and HxWO3 have applications as catalysts and in electrochromic displays, with the proton sites and motion being central to their properties. This paper describes the use of muon spin relaxation (mu SR), where the positive muon mimics protons, to investigate the proton behaviour in the parent oxide ReO3. This was chosen for our initial study as it has a simple cubic structure (the W and Mo analogues having a distorted version), and the Re nucleus has a suitable nuclear moment to interact with the muons. Spin-polarised positive muons are implanted into the sample, and the evolution of the polarization over their lifetime is measured. Muons can behave as a light isotope of hydrogen as they have the same charge as the proton, but with a mass of 1/9 of that of the proton, there is a large isotope effect leading to larger zero-point motion and more rapid diffusion. Interaction between the muon's spin and nearby nuclear magnetic moments causes the muon to depolarise. Analysis of the zero-field muon spin relaxation shows that the muon is at an interstitial site for temperatures around 50-100 K. The relaxation rate is consistent with the muon being bonded to an oxide ion, forming the analogue of OH-, which is then freely rotating about the Re-O-Re axis. Below 25 K, the relaxation rate increases, suggesting a slowing down of this rotation. Around 150-200 K, the muon is hopping between sites, and in the range 300-400 K, another stable site is reached. This has a lower relaxation rate indicating that it is farther from the nearest Re ions, and is probably near a defect in the crystal structure such as Re vacancy.