In this work, we introduce Conducting Atomic Force Microscopy (C-AFM) as a novel technique for the determination of the local effective electrical oxide thickness with a lateral resolution of a few nanometers and a thickness resolution in the sub ångström range. In this technique the conductive tip of an AFM, which is in mechanical contact with the bare oxide surface, is used as metal electrode to define a local MOS structure with nanometer lateral extension. Oxide thickness determination is done by fitting the local I-V curves to the well known Fowler Nordheim tunneling equation with a thickness sensitivity in the sub-ångström range. In addition, tunneling current images at constant applied voltage can be obtained simultaneously to the oxide surface topography. We present a scheme which allows the conversion of the tunneling current images into maps of the local electrical oxide thickness. Several examples demonstrate the versatile and far-reaching application of C-AFM to R&D and failure analysis.