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Abstract. A permutomino of size n is a polyomino whose vertices define a pair of distinct permutations of length n. In this paper we treat various classes of convex permutominoes, including the parallelogram, the directed convex and the stack ones. Using bijective techniques we provide enumeration for each of these classes according to the size, and characterize the permutations which are associated with permutominoes of each class. Mathematics Subject Classifications (2000). 05A15, 05A05