We investigate QCD thermodynamics at finite quark chemical potential. Lattice QCD results are compared with a generalized Nambu Jona-Lasinio model in which quarks couple simultaneously to the chiral condensate and to a background temporal gauge field representing Polyakov loop dynamics. This so-called PNJL model thus includes features of both deconfinement and chiral symmetry restoration. The equation of state of a system of two- and two plus one- quark flavours is evaluated, and the results are compared to the corresponding lattice data, at zero and finite quark chemical potential. The phase diagram of the model, in the temperature-chemical potential plane, is obtained, and its quark-mass dependence is investigated. Predictions are made for the phase diagram at values of the physical quark masses, still inaccessible by the present lattice calculations. The position of the tricritical point, separating first-order phase transition from crossover, is studied as a function of the bare quark masses. Regions of high chemical potential are explored in the two-flavour case, and the possibility of diquark condensation and colour superconductivity is taken into account.