The European-Mediterranean earthquake catalogue from 1901 to 1985, which comprises uniformly determined magnitudes M(s) and m(B)(h greater-than-or-equal-to 60 km) of 13300 events, was used in the study of cumulative magnitude-frequency relationships N(c)(M) compiled for 75 earthquake regions and 25 larger provinces. In the whole magnitude range observed, the Gutenberg-Richter formula log N(c)(M) = a - bM very rarely fits the cumulative (log N(c), M) distributions. The b-values of log-linear segments of N(c)(M) vary regionally from b = 0.7 to b = 1.3; averaging of all values leads to bBAR = 0.92 (shallow events, M(s)) and bBAR = 0.56 (h greater-than-or-equal-to 60 km, m(B)). Most distributions pertain to the Mediterranean area (b = 0.86 from the graph for shallow events) and many of them indicate the existence of characteristic earthquakes in accordance with the theoretical single-fault model. Other observed shapes of N(c)(M) can be explained by the superposition of populations of different M(max) values or by the presence of swarm-type activity. The observed N(c)(M) distributions depend very much on the delineation of earthquake regions, i.e. on the number and dimension of seismoactive faults in the investigated region. A premonitory enhancement of medium earthquake activity (M = 4.5-5.5) can be observed only very rarely.