Aldo Procacci, Benedetto Scoppola On decay of correlations for unbounded spin systems with arbitrary boundary conditions. (67K, plain tex) ABSTRACT. We propose a method based on cluster expansion to study the truncated correlations of unbounded spin systems uniformly in the boundary condition and in a possible external field. By this method we study the spin-spin truncated correlations of various systems, including the case of infinite range simply integrable interactions, and we show how suitable boundary conditions and/or external fields may improve the decay of the correlations.