AbstractAny height system has two constituents: A reference surface upon which all heights are equal to zero, and a prescription for how observed heights and height differences will be related to that surface. That prescription is typically formulated with reference to Earth’s gravity field, but in this contribution, we will use the concept of metric spaces instead. In most height systems, the height of a point can be interpreted as the length of the 3-dimensional path from a point of interest to the reference surface in a particular metric space. The geometry of the path is that of the space associated with the height system. This submission explores the definition of a height system simply as a metric space and a reference surface, applies it to common height systems used in geodesy (geodetic, orthometric, dynamic, normal), and examines their characteristics through that lens.