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Recent developments of discrete methods of fluid dynamics, particularly the two most relevant aspects: the half discrete case discrete Boltzmann equation; and the totally discrete one lattice gas were discussed. Both the conceptual and numerical significance of these discrete models were covered as well as the mathematical problems which arise from them. This Colloquium is the third of a series initiated in Santa Fe (USA 1986) the second having taken place in Torino (Italy 1988).