We have used X-ray absorption near edge structure (XANES) as a probe of the local atomic structure of titanate perovskites. In their low-temperature phases, most titanate perovskites are distorted from the ideal perovskite structure and this distortion is evident in the XANES. We have measured the XANES of several materials, ATiO(3), where A is an element of {Pb, Ba, Sr, Eu} and of several materials, PbTixZr1-xO3 where x is an element of {1.0, 0.52, 0.3}. In the strongly distorted pure materials, PbTiO3 and BaTiO3, we seen features a few eV below the main edge that are indicative of a 1s --> 3d electronic transition. This normally quadrupole transition is enhanced by the addition of a dipole transition from the mixture of p character into the 3d final state due to the broken centrosymmetry of the crystal. In SrTiO3, and EuTiO3, where the distortion is minuscule or absent, the lowest-energy pre-edge feature becomes much smaller. We also measured the XANES of the PbTixZr1-xO3 materials as a function of temperature. We found the pre-edge features of these to diminish somewhat in the paraelectric phase, but remain much stronger than in the undistorted EuTiO3. The nature of this pre-edge feature is mostly independent of the titanium concentration x suggesting that local structure about the titanium site is only weakly dependent on crystallographic phase. In PbTiO3 this is consistent with the order-disorder character of its ferroelectric transition. The cause of this behavior in the pre-edge features is discussed.