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We consider two-dimensional error-correcting codes capable of correcting a single arbitrary cluster of errors of size b. We provide optimal 2-cluster-correcting codes in several connectivity models, as well as optimal, or nearly optimal, 2-cluster-correcting codes in all dimensions. We also construct 3-cluster-correcting codes and b-straight-cluster-correcting codes. We conclude by improving the Reiger bound for two-dimensional cluster-correcting codes.