Most metal-semiconductor contacts are rectifying. For moderately doped semiconductors, the current transport across such Schottky contacts occurs by thermionic emission over the Schottky barrier. The current-voltage characteristics of real Schottky contacts are described by two fitting parameters that are the effective barrier heights Phi(B)(eff) and the ideality factors n. Due to lateral inhomogeneities of the barrier height, both parameters differ from one diode to another. However, their variations are correlated in that Phi(B)(eff) becomes smaller with increasing n. Extrapolations of such Phi(B)(eff)-versus-n plots to the corresponding image-force-controlled ideality factors nif give the barrier heights of laterally homogeneous contacts. They are then compared with the theoretical predictions for ideal Schottky contacts. Data of Si, GaN, GaAs, and CdTe Schottky contacts reveal that the continuum of metal-induced gap states is the fundamental mechanism that determines the barrier heights. However, there are additional but then secondary mechanisms. As an example, contacts with (7X7)(i)-reconstructed interfaces have smaller barrier heights than diodes with (1X1)(i)-unreconstructed interfaces. This lowering of the Schottky barrier is caused by the electric dipole associated with the stacking fault in one of the triangular halves of the (7 X 7) unit mesh.