KMOS is a multi-object near-infrared integral field spectrometer with 24 deployable cryogenic pick-off arms. Inevitably, data processing is a complex task that requires careful calibration and quality control. In this paper we describe all the steps involved in producing science-quality data products from the raw observations. In particular, we focus on the following issues: (i) the calibration scheme which produces maps of the spatial and spectral locations of all illuminated pixels on the detectors; (ii) our concept of minimising the number of interpolations, to the limiting case of a single reconstruction that simultaneously uses raw data from multiple exposures; (iii) a comparison of the various interpolation methods implemented, and an assessment of the performance of true 3D interpolation schemes; (iv) the way in which instrumental flexure is measured and compensated. We finish by presenting some examples of data processed using the pipeline. ; Comment: accepted for publication in A&A