The detection of methane on Mars has been interpreted as indicating that geochemical or biotic activities could persist on Mars today(1). A number of different measurements of methane show evidence of transient, locally elevated methane concentrations and seasonal variations in background methane concentrations(2-5). These measurements, however, are difficult to reconcile with our current understanding of the chemistry and physics of the Martian atmosphere(6,7), which-given methane's lifetime of several centuries-predicts an even, well mixed distribution of methane(1,6,8). Here we report highly sensitive measurements of the atmosphere of Mars in an attempt to detect methane, using the ACS and NOMAD instruments onboard the ESA-Roscosmos ExoMars Trace Gas Orbiter from April to August 2018. We did not detect any methane over a range of latitudes in both hemispheres, obtaining an upper limit for methane of about 0.05 parts per billion by volume, which is 10 to 100 times lower than previously reported positive detections(2,4). We suggest that reconciliation between the present findings and the background methane concentrations found in the Gale crater(4) would require an unknown process that can rapidly remove or sequester methane from the lower atmosphere before it spreads globally.