Two observational campaigns were carried out during the eclipses of EE Cep in 2003 and 2008/9 to verify whether the eclipsing body in the system is indeed a dark disk and to understand the observed changes in the depth and durations of the eclipses. Multicolour photometric data and spectroscopic observations at both low and high resolution were collected. We numerically modelled the variations in brightness and colour during the eclipses. We tested models with different disk structure. We considered the possibility of disk precession. The complete set of observational data collected during the last three eclipses are made available to the astronomical community. Two blue maxima in the colour indices were detected during these two eclipses, one before and one after the photometric minimum. The first (stronger) blue maximum is simultaneous with a "bump" that is very clear in all the UBVRI light curves. Variations in the spectral line profiles seem to be recurrent during each cycle. NaI lines always show at least three absorption components during the eclipse minimum and strong absorption is superimposed on the H_alpha emission. These observations confirm that the eclipsing object in EE Cep system is indeed a dark, dusty disk around a low luminosity object. The primary appears to be a rapidly rotating Be star that is strongly darkened at the equator and brightened at the poles. Some of the conclusions of this work require verification in future studies: (i) a complex, possibly multi-ring structure of the disk in EE Cep; (ii) our explanation of the "bump" observed during the last two eclipses in terms of the different times of obscuration of the hot polar regions of the Be star by the disk; and (iii) our suggested period of the disk precession (~11-12 P_orb) and predicted depth of about 2 mag the forthcoming eclipse in 2014. ; Comment: 43 pages, 17 figures, accepted for publication in A&A