We establish some results on the Wadge degrees and on the Boolean hierarchy of k-partitions of some spaces, where k is a natural number. The main attention is paid to the Baire space, Baire domain and their close relatives. For the case of mathbfDelta20-measurable k-partitions the structures of Wadge degrees are characterized completely. For many degree structures, undecidability of the first-order theories is shown, for any k ≥ 3. © 2009 Elsevier Inc. All rights reserved.