Context. The Medium Resolution integral field Spectrometer (MRS) of the Mid-Infrared Instrument (MIRI) on board the James Webb Space Telescope (JWST) performs spectroscopy between 5 and 28 µm, with a field of view varying from ~13 to ~56 sq. arcsec. The optics of the MRS introduce substantial distortion and this needs to be rectified in order to reconstruct the observed astrophysical scenario. Aims. We aim to use data from the JWST/MIRI commissioning and cycle 1 calibration phase to derive the MRS geometric distortion and astrometric solution, a critical step in the calibration of MRS data. These solutions come in the form of transform matrices that map the detector pixels to spatial coordinates of a local MRS coordinate system called α/β, to the global JWST observatory coordinates V2 and V3 (V2+V3). Methods. For every MRS spectral band and each slice dispersed on the detector, we fit the transform of detector pixels to α/β by a two-dimensional (2D) polynomial, using a raster of point source observations. The dispersed trace of the point source on the detector was initially estimated by fitting a one-dimensional (1D) empirical function and then iterating on the first distortion solution using forward modelling of the point spread function model based on the webbpsf python package. A polynomial transform was used to map the coordinates from α/β to V2+V3. Results. We calibrated the distortion of all 198 discrete slices of the MIRI/MRS integral field units and derived an updated field of view (FoV) for each MRS spectral band. The precision of the distortion solution is estimated to be better than one-tenth of a spatial resolution element, with a root mean square (rms) of 10 milli-arc-second (mas) at 5 µm, to 23 mas at 27 µm. Finally, we found that the wheel positioning repeatability causes an additional astrometric rms error of 30 mas. Conclusions. We demonstrate the application of the MRS astrometric calibration strategy and analysis for all four integral field units and all spectral bands of the MRS that enable the calibration of MRS spectra. This is a critical step in the data pipeline of every MRS observation. The distortion calibration was folded into the JWST pipeline in the Calibration Reference Data System (CRDS) context (jwst_1094.pmap), meeting the pre-launch requirement, with an estimated total astrometric uncertainty of 50 mas.