First results are presented of an effort to model the storm-time distortion of the magnetic field in the inner magnetosphere using space magnetometer data. Strong geomagnetic storms are relatively rare events, represented by only a small fraction of the data used in the derivation of existing empirical geomagnetic field models. Hence using those models for the mapping of the storm-time magnetosphere is at most an extrapolation based on trends, obtained from quiet and moderately disturbed data. To overcome that limitation, a set of data was created, containing only clear-cut events with Dst ≤ -65 nT, with the goal to derive models of the inner and near geomagnetic field (R < 15 RE), representing strongly disturbed geomagnetic configurations and their evolution during the storm cycle. The final data set included about 143,000 records with 5-min average B-vectors, covering 37 major storms between 1996 and 2000. Most of the data came from GOES-8, -9, -10, Polar, and Geotail spacecraft, and two storms in February--March of 1998 were also partially covered by the data of Equator-S. In all cases, only those storms were selected for which concurrent solar wind and IMF data were available for the entire duration of the event. Interplanetary medium data were provided by Wind, ACE, and, to a lesser extent, by IMP 8 and Geotail. The inner magnetospheric field was represented using the newly developed T01 model <Tsyganenko, 2002a, 2002b>, with a duskside partial ring current with variable amplitude and scale size, an essential part of the storm-time current system. The modeling revealed an enormous distortion and dawn-dusk asymmetry of the inner magnetosphere during the peak of the storm main phase, caused by the combined effect of the symmetric and partial ring currents, cross-tail current, and Birkeland currents. We found that during storms with Dst < -250 nT the tail-like deformation of the nightside field penetrates so close to Earth that the quasi-dipolar approximation breaks down at distances as small as 3--4 RE. This finding yields a quantitative answer to the question of why the auroras expand to unusually low latitudes during extremely strong storms. It also may provide a natural explanation for the observed impulsive injections and energizations of charged particles on the innermost L-shells. Finally, it questions the validity of using the dipole or quasi-dipole approximation in numerical simulations of severe storms in the inner magnetosphere.