During the last week of May 2003, the solar active region AR 10365 produced a large number of flares, several of which were accompanied by Coronal Mass Ejections (CME). Specifically on 27 and 28 May three halo CMEs were observed which had a significant impact on geospace. On 29 May, upon their arrival at the L1 point, in front of the Earth's magnetosphere, two interplanetary shocks and two additional solar wind pressure pulses were recorded by the ACE spacecraft. The interplanetary magnetic field data showed the clear signature of a magnetic cloud passing ACE. In the wake of the successive increases in solar wind pressure, the magnetosphere became strongly compressed and the sub-solar magnetopause moved inside five Earth radii. At low altitudes the increased energy input to the magnetosphere was responsible for a substantial enhancement of Region-1 field-aligned currents. The ionospheric Hall currents also intensified and the entire high-latitude current system moved equatorward by about 10°. Several substorms occurred during this period, some of them - but not all - apparently triggered by the solar wind pressure pulses. The storm's most notable consequences on geospace, including space weather effects, were (1) the expansion of the auroral oval, and aurorae seen at mid latitudes, (2) the significant modification of the total electron content in the sunlight high-latitude ionosphere, (3) the perturbation of radio-wave propagation manifested by HF blackouts and increased GPS signal scintillation, and (4) the heating of the thermosphere, causing increased satellite drag. We discuss the reasons why the May 2003 storm is less intense than the October-November 2003 storms, although several indicators reach similar intensities.