Perovskites of the PbZr1-xTixO3 type are among the most important ferroelectric materials and highly active catalysts. The structural and electronic properties of PbTiO3, PbZrO3, and PbZr0.5Ti0.5O3 were examined using first-principles density-functional (DF) calculations with the local-density-approximation (LDA) or the generalized-gradient approximation (GGA, Perdew-Wang and Perdew-Burke-Ernzerhoff functionals). A series of crystal structures were considered for each compound. In several cases, the structural parameters predicted by the GGA functionals were clearly in better agreement with experimental results than the LDA-predicted values, but in qualitative terms the LDA and GGA approaches always predicted similar trends for crystal geometries and differences in thermochemical stability. DF calculations at the LDA level could underestimate the ferroelectric character of PbTiO3 and PbZr1-xTixO3. In the perovskites, the most stable structures belong to tetragonal (PbTiO3), orthorhombic (PbZrO3), and monoclinic (PbZr0.5Ti0.5O3) space groups. The positions of the Zr and Ti cations in the tetragonal and monoclinic phases of PbZr0.5Ti0.5O3 were determined. The calculated structural parameters give theoretical x-ray diffraction patterns that reproduce well experimental data. In general, Zr is much more rigid than Ti for displacements along the [001] direction in the cubic, tetragonal and monoclinic phases of PbZr1-xTixO3 compounds. The lead titanates/zirconates exhibit very strong metal<---->oxygen<---->metal interactions that drastically modify the electron density on the metal cations with respect to TiO2, ZrO2, and PbO. A similar phenomenon is observed in a series of ABO(3) perovskites (A=Ca,Sr,Li,K,Na; B=Ti,Zr,Nb), and it is an important factor to consider when mixing AO and BO2 oxides for catalytic applications.