Hexagonal boron nitride (h-BN) is a promising two-dimensional insulator with a large band gap and low density of charged impurities that is isostructural and isoelectronic with graphene. Here we report the chemical and atomic-scale structure of CVD-grown wafer-scale (~25 cm2) h-BN sheets ranging in thickness from 1-20 monolayers. Atomic-scale images of h-BN on Au and graphene/Au substrates obtained by scanning tunneling microscopy (STM) reveal high h-BN crystalline quality in monolayer samples. Further characterization of 1-20 monolayer samples indicates uniform thickness for wafer-scale areas; this thickness control is a result of precise control of the precursor flow rate, deposition temperature and pressure. Raman and infrared spectroscopy indicate the presence of B-N bonds and reveal a linear dependence of thickness with growth time. X-ray photoelectron spectroscopy (XPS) shows the film stoichiometry, and the B/N atom ratio in our films is 1 ± 0.6% across the range of thicknesses. Electrical current transport in metal/insulator/metal (Au/h-BN/Au) heterostructures indicates that our CVD-grown h-BN films can act as excellent tunnel barriers with a high hard-breakdown field strength. Our results suggest that large-area h-BN films are structurally, chemically and electronically uniform over the wafer scale, opening the door to pervasive application as a dielectric in layered nanoelectronic and nanophotonic heterostructures.