The method of linearized delay-time tomography is applied to retrieve the P velocity heterogeneity of the upper mantle in central Europe, the Mediterranean and Asia Minor. The upper mantle is parametrized in slowness cell-functions with lateral dimensions of 1d̀. The model parametrization includes event mislocation vectors and station delay-time corrections. In all 20 000 parameters describe the model. Delay-time data, selected from the ISC data base (1964-1982), from regional events and from stations up to 90d̀ of epicentral distance are used. About 18 000 events and 937 stations contributed nearly 500 000 delay times. We discuss the tomographic method, the data processing, the cell model employed, the computation of the ray geometry, the use of composite rays in tomography, and the cell hit count. The inversions are performed with two different iterative row-action methods, LSQR and SIRT. Results obtained with the LSQR method will be shown only. The spatial resolution is investigated with sensitivity analyses using two different synthetic velocity models; a ‘spike’ model and a model with harmonically varying velocity anomalies. The spatial resolution is estimated to be high (on the order of the cell size) in the mantle below central and southeastern Europe. The mapped velocity heterogeneity exhibits primarily low velocities around a depth of 200 km and just above the transition from upper to lower mantle. This indicates a systematic deviation of the reference model (Jeffreys-Bullen) from the radially averaged earth structure, which may violate the basic assumptions underlying the linearized approach. A posteriori investigation of the data also indicates the presence of a low-velocity zone and the existence of strong velocity depth gradients.