Magnetization curves of square arrays of Josephson junctions of two basic types were investigated: superconductor-insulator-superconductor (SIS) and superconductor-normal metal-superconductor (SNS). Magnetic flux avalanches were observed in SIS arrays. A statistical analysis of flux avalanches showed that their size distribution can be described by a power law with a crossover where the exponent n varies from -1.2 for small avalanches to -3.5 for the large ones. Such a behavior of avalanches is interpreted as the self-organized criticality (SOC) manifestation. In SNS arrays, the flux avalanches were not observed, but a considerable asymmetry of a hysteresis curve was revealed. (c) 2004 Elsevier B.V. All rights reserved.