A scanning tunneling microscope is used to demonstrate the principle of quantum mechanical tunneling between the microscope tip and the surface of a conducting sample. Measurements are made on a gold-coated holographic grating and a pyrolytic graphite sample. Since the apparatus is capable of atomic resolution, atomic features of the graphite surface can be directly observed. Mathematical filter algorithms are used to process the sample images and reduce the image noise. The bond angles and bond lengths of the graphite sample are determined.