Extended optical and near-IR observations reveal that SN 2009dc shares a number of similarities with normal Type Ia supernovae (SNe Ia), but is clearly overluminous, with a (pseudo-bolometric) peak luminosity of log (L) = 43.47 (erg s-1). Its light curves decline slowly over half a year after maximum light [delta m(15)(B)(true) = 0.71], and the early-time near-IR light curves show secondary maxima, although the minima between the first and the second peaks are not very pronounced. The bluer bands exhibit an enhanced fading after similar to 200 d, which might be caused by dust formation or an unexpectedly early IR catastrophe. The spectra of SN 2009dc are dominated by intermediate-mass elements and unburned material at early times, and by iron-group elements at late phases. Strong C ii lines are present until similar to 2 weeks past maximum, which is unprecedented in thermonuclear SNe. The ejecta velocities are significantly lower than in normal and even subluminous SNe Ia. No signatures of interaction with a circumstellar medium (CSM) are found in the spectra. Assuming that the light curves are powered by radioactive decay, analytic modelling suggests that SN 2009dc produced similar to 1.8 M-circle dot of 56Ni assuming the smallest possible rise time of 22 d. Together with a derived total ejecta mass of similar to 2.8 M-circle dot, this confirms that SN 2009dc is a member of the class of possible super-Chandrasekhar-mass SNe Ia similar to SNe 2003fg, 2006gz and 2007if. A study of the hosts of SN 2009dc and other superluminous SNe Ia reveals a tendency of these SNe to explode in low-mass galaxies. A low metallicity of the progenitor may therefore be an important prerequisite for producing superluminous SNe Ia. We discuss a number of possible explosion scenarios, ranging from super-Chandrasekhar-mass white-dwarf progenitors over dynamical white-dwarf mergers and Type I<file name="mnr_18107_mu1.gif" type="gif"/> SNe to a core-collapse origin of the explosion. None of the models seems capable of explaining all properties of SN 2009dc, so that the true nature of this SN and its peers remains nebulous.