Abstract The usual classification of phase transitions does not allow for anomalous transitions, such as jumps in a pressure isotherm at constant density, which appear in certain statistical mechanical models of fluids and spin systems (which embody many-body interactions of indefinitely high order). These experimentally unobserved but thermodynamically allowed phase transitions may appear as discontinuities in any first-order derivative of any thermodynamic function other than the appropriate Gibbs free energy. First-order phase transitions, including the anomalous ones, may be grouped into distinct types by identifying six classes of discontinuity in the graphs of y versus x at constant z , where x, y and z can be chosen in 24 ways from a set of two thermodynamic densities, Q a and Q b (or, loosely, “extensive” variables), such as the entropy density and particle density, and their conjugate thermodynamic fields, h a and h b , such as the temperature, T , and chemical potential μ. Thermodynamic convexity implies that class 2 discontinuities, typified by ( x , y , z ) = ( h a , h b , Q b ), may occur only at isolated values of z . In class 4, typified by ( x , y , z ) = ( Q a , h a , h b ) the value of x , say x Δ ( z ), at which the discontinuity occurs must be piecewise constant ; conversely, in class 5, typified by ( x , y , z ) = ( h a , Q b , h b ), x Δ ( z ) cannot remain fixed over any nonvanishing range of z . A jump in the pressure versus density, ϱ, isotherm is in class 4 and so the transition point, ϱ Δ ( T ), must remain fixed as T varies locally or, more generally, must be a piecewise constant function of T . The allowed discontinuities fall into twenty distinct groups of thermodynamically associated discontinuities , any particular discontinuity appearing in two or, sometimes, three groups. The twenty groups are, in turn, generated by nine distinct transition types which specify the geometrical structure of the internal energy (or entropy) surface and of the conjugate Gibbs free energy surface, indicating ridges and ruled surfaces, aligned ridges and cylinders, peaks and planar facets, generic or aligned linear ridges, generic or aligned pinch spots and isolated ruled lines, aligned isotransclines, and isosectional surfaces. All thermodynamically allowed discontinuities and transition types, normal or anomalous, can be realized in exactly soluble models with suitable Hamiltonians.