In this paper, we present the first magnetic images of the rapidly rotating young K0 dwarf AB Doradus. Simultaneous brightness images are also reconstructed. These images are obtained from two independent data sets recorded four to six nights (eight to twelve rotation cycles) apart, with the UCL Echelle Spectrograph of the Anglo-Australian Telescope in 1995 December. All images are recovered from sets of `average profiles' obtained through a least-squares deconvolution process of about 1500 lines throughout the whole visible spectrum. We find that about 20 per cent of the surface of AB Dor is covered with a significant (>200G) field, whose flux within these magnetic regions is typically 500 G, with peaks up to 1.5 kG. The associated magnetic field topology is essentially radial (in 1995 December) and features an alternating east-west polarity structure at intermediate to high latitudes, comprising up to 12 regions of opposite polarities. We suspect that this pattern represents the emergence of buoyant flux tubes from an underlying toroidal field structure. The toroidal field itself is also detected at the photospheric level, and switches from positive polarity at intermediate latitudes to negative polarity in circumpolar regions. Altogether, it suggests that the associated dynamo is probably not purely solar-like (i.e. confined to an overshoot layer at the base of the convective zone), but must also feature a significant component distributed throughout the whole convective envelope. From the evolution of brightness maps throughout eight rotations, we infer that the pole of AB Dor rotates more slowly than the equator, by about one part in 220. The corresponding `lap time' required for the equatorial region to lap the pole is thus of the order of 110d, very close to the solar value of 120d. Along with the information that the photometric period has been in steady decrease since 1988, our result implies that cool features have moved equatorward on average on AB Dor. Repeated observations of the Hα-absorbing circumstellar prominence system indicate rotation periods for individual prominences that are significantly longer than those obtained for the mid-latitude magnetic features. We infer that if the prominences corotate with the footpoints of the magnetic structures in which they form, they must be anchored at latitudes greater than 60 deg.