We studied the temporal evolution of fractality for geomagnetic activity, by calculating fractal dimensions from the Dst data and from a magnetohydrodynamic shell model for turbulent magnetized plasma, which may be a useful model to study geomagnetic activity under solar wind forcing. We show that the shell model is able to reproduce the relationship between the fractal dimension and the occurrence of dissipative events, but only in a certain region of viscosity and resistivity values. We also present preliminary results of the application of these ideas to the study of the magnetic field time series in the solar wind during magnetic clouds, which suggest that it is possible, by means of the fractal dimension, to characterize the complexity of the magnetic cloud structure.