Numerical notation systems are graphic systems for representing numbers, such as the Roman and Hindu-Arabic numerals. Historians of mathematics have traditionally classified these systems on the basis of only one criterion: whether or not they use place-value (positional notation) to express numbers. Such typologies inadequately represent the various structural features of systems, and place excessive weight on positionality as the only relevant or interesting issue. In their place, I propose a two-dimensional typology that better reflects the variety of systems encountered historically. This new typology emphasizes that all numerical notation systems have advantages and disadvantages, and allows interesting historical questions to be posed regarding the evolution of numerical notation from antiquity to the present.