Detailed studies of the properties of ceramic CaCu3Ti4O12 (CCTO) have clarified the physics of this interesting material and revealed several features not reported before. The dielectric relaxational properties of CCTO are explained in terms of a capacitive-layer model, as for an inhomogeneous semiconductor, consisting of semiconducting grains and insulating grain boundaries as also concluded by others. The kinetics of the main [low-temperature (T)] relaxation reveal that two different thermally activated processes in CCTO grains control the dynamics. A likely candidate defect responsible for the two processes is the oxygen vacancy which is a double donor. A higher-T relaxation is determined by grain boundary conduction. Both Nb and Fe doping lowered both the apparent dielectric constant epsilon(') and the dielectric loss, but increased Fe doping led to more dramatic effects. At 3 at. % Fe doping, the anomalous epsilon(')(T) response was removed, making the CCTO an intrinsic, very-low-loss dielectric. The intrinsic epsilon(')(similar or equal to 75) and its T dependence are measured and shown to be largely determined by a low-lying soft TO phonon. At low T, cubic CCTO transforms into an antiferromagnetic phase at T-N=25 K. T-N is essentially independent of Nb doping (up to 4 at. %) and of hydrostatic pressure (up to similar to 7 kbar), but decreases significantly with Fe doping. Analysis of the high-T dependence of the magnetic susceptibility provided insight into the role of Fe as a dopant. Finally, an epsilon(')(T) anomaly associated with the onset of antiferromagnetic order has been discovered, providing evidence for coupling between the polarization and sublattice magnetization. The possible origin of this coupling is discussed.