Author(s) 2019. Carbon monoxide (CO) is an important atmospheric constituent affecting air quality, and methane (CH4) is the second most important greenhouse gas contributing to human-induced climate change. Detailed and continuous observations of these gases are necessary to better assess their impact on climate and atmospheric pollution. While surface and airborne measurements are able to accurately determine atmospheric abundances on local scales, global coverage can only be achieved using satellite instruments. The TROPOspheric Monitoring Instrument (TROPOMI) onboard the Sentinel-5 Precursor satellite, which was successfully launched in October 2017, is a spaceborne nadirviewing imaging spectrometer measuring solar radiation reflected by the Earth in a push-broom configuration. It has a wide swath on the terrestrial surface and covers wavelength bands between the ultraviolet (UV) and the shortwave infrared (SWIR), combining a high spatial resolution with daily global coverage. These characteristics enable the determination of both gases with an unprecedented level of detail on a global scale, introducing new areas of application. Abundances of the atmospheric column-averaged dry air mole fractions XCO and XCH4 are simultaneously retrieved from TROPOMI's radiance measurements in the 2:3 μm spectral range of the SWIR part of the solar spectrum using the scientific retrieval algorithm Weighting Function Modified Differential Optical Absorption Spectroscopy (WFMDOAS). This algorithm is intended to be used with the operational algorithms for mutual verification and to provide new geophysical insights. We introduce the algorithm in detail, including expected error characteristics based on synthetic data, a machine-learning-based quality filter, and a shallow learning calibration procedure applied in the post-processing of the XCH4 data. The quality of the results based on real TROPOMI data is assessed by validation with ground-based Fourier transform spectrometer (FTS) measurements providing realistic error estimates of the satellite data: the XCO data set is characterised by a random error of 5:1 ppb (5:8 %) and a systematic error of 1:9 ppb (2:1 %); the XCH4 data set exhibits a random error of 14:0 ppb (0:8 %) and a systematic error of 4:3 ppb (0:2 %). The natural XCO and XCH4 variations are well-captured by the satellite retrievals, which is demonstrated by a high correlation with the validation data (R = 0:97 for XCO and R = 0:91 for XCH4 based on daily averages). We also present selected results from the mission start until the end of 2018, including a first comparison to the operational products and examples of the detection of emission sources in a single satellite overpass, such as CO emissions from the steel industry and CH4 emissions from the energy sector, which potentially allows for the advance of emission monitoring and air quality assessments to an entirely new level.