Several classes of finite distance spaces and the corresponding classes of graphs are considered \((l_ 1\)-space, \(h\)-embeddable, spherical, hypermetric and of negative type). The focus is on hypermetric graphs. Several characterization results for hypermetric graphs are given. In particular, connected regular hypermetric graphs of diameter \(\leq 2\) are characterized and hypermetric strongly regular graph are enumerated. Hypermetric distance regular graphs are also investigated. Characterization of hypermetric Taylor graphs is given. It is also shown that for Taylor graphs it is equivalent to be hypermetric, to be of negative type, and to be spherical. Characterizations of extreme and regular hypermetric graphs are also derived.