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Context. EE Cep is an unusual long-period (5.6 yr) eclipsing binary discovered during the mid-twentieth century. It undergoes almost-grey eclipses that vary in terms of both depth and duration at different epochs. The system consists of a Be type star and a dark dusty disk around an invisible companion. EE Cep together with the widely studied ε Aur are the only two known cases of long-period eclipsing binaries with a dark, dusty disk component responsible for periodic obscurations. Aims. 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 depths and durations of the eclipses. Methods. Multicolour photometric data and spectroscopic observations performed at both low and high resolutions were collected with several dozen instruments located in Europe and North America. We numerically modelled the variations in brightness and colour during the eclipses. We tested models with different disk structure, taking into consideration the inhomogeneous surface brightness of the Be star. We considered the possibility of disk precession. Results. The complete set of observational data collected during the last three eclipses are made available to the astronomical community. The 2003 and 2008/9 eclipses of EE Cep were very shallow. The latter is the shallowest among all observed. The very high quality photometric data illustrate in detail the colour evolution during the eclipses for the first time. 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 UBV(RI)C light curves. A temporary increase in the I-band brightness at the orbital phase ∼0.2 was observed after each of the last three eclipses. Variations in the spectral line profiles seem to be recurrent during each cycle. The Na i lines always show at least three absorption components during the eclipse minimum and strong absorption is superimposed on the Hα emission. Conclusions. 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 Porb) and predicted depth of about 2m ̇ for the forthcoming eclipse in 2014. © 2012 ESO.