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We studied the effects of bulk and surface conductivity on the performance of high-resistivity CdZnTe (CZT) pixel detectors with Pt contacts. We emphasize the difference in mechanisms of the bulk and surface conductivity as indicated by their different temperature behaviors. In addition, the existence of a thin (10-100 Å) oxide layer on the surface of CZT, formed during the fabrication process, affects both bulk and surface leakage currents. We demonstrate that the measured I-V dependencies of bulk current can be explained by considering the CZT detector as a metal-semiconductor-metal system with two back-to-back Schottky-barrier contacts. The high-surface leakage current is apparently due to the presence of a low-resistivity surface layer that has characteristics that differ considerably from those of the bulk material. This surface layer has a profound effect on the charge-collection efficiency in detectors with multicontact geometry; some fraction of the electric field lines that originated on the cathode intersects the surface areas between the pixel contacts where the charge produced by an ionizing particle gets trapped. To overcome this effect, we place a grid of thin electrodes between the pixel contacts. When the grid is negatively biased, the strong electric field in the gaps between the pixels forces the electrons landing on the surface to move toward the contacts, preventing the charge loss. We have investigated these effects by using CZT pixel detectors indium bump-bonded to a custom-built VLSI readout chip.