This article reviews the use of scanning tunneling microscopy (STM) and scanning tunneling spectroscopy (STS) to characterize the physical and electronic properties of epitaxial graphene. Topographical variations revealed by STM allow the determination of the number of graphene layers and the detection of lattice mismatch between the graphene and the substrate, as well as rotational disorder. STS allows the local electronic characterization of graphene. STM/STS can also be used to perform local studies of graphene modification through processes such as atomic/molecular adsorption and intercalation.