The acquisition of new global elevation data from the Lunar Orbiter Laser Altimeter, carried on the Lunar Reconnaissance Orbiter, permits quantification of the surface roughness properties of the Moon at unprecedented scales and resolution. We map lunar surface roughness using a range of parameters: median absolute slope, both directional (along-track) and bidirectional (in two dimensions); median differential slope; and Hurst exponent, over baselines ranging from ~17 m to ~2.7 km. We find that the lunar highlands and the mare plains show vastly different roughness properties, with subtler variations within mare and highlands. Most of the surface exhibits fractal-like behavior, with a single or two different Hurst exponents over the given baseline range; when a transition exists, it typically occurs near the 1 km baseline, indicating a significant characteristic spatial scale for competing surface processes. The Hurst exponent is high within the lunar highlands, with a median value of 0.95, and lower in the maria (with a median value of 0.76). The median differential slope is a powerful tool for discriminating between roughness units and is useful in characterizing, among other things, the ejecta surrounding large basins, particularly Orientale, as well as the ray systems surrounding young, Copernican-age craters. In addition, it allows a quantitative exploration on mare surfaces of the evolution of surface roughness with age.