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In this work, we investigate the advantages and challenges of deploying a single mobile sink in IEEE 802.15.4/ZigBee wireless sensor networks (WSNs). The first part of the paper provides an overview of the most recent research on sink mobility in WSNs, placing a special emphasis on different types of sink mobility (random, predictable and controlled) and discussing the application scenarios most suitable for their respective deployment. In the second part of the paper, our OPNET model for simulation of large-scale and ZigBee-based wireless sensor networks is presented. The model enables effective evaluation of random and predictable sink mobility under varying conditions and forms of routing in the underlying ZigBee WSN. The results obtained using this model show that in terms of energy efficiency ZigBeepsilas tree-based routing outperforms ZigBeepsilas mesh routing, both in the case of random and predictable sink mobility. At the same time, under both mobility models, tree-based routing generates longer delays in the delivery of data reporting packets. Furthermore, when compared against each other assuming identical network conditions, random mobility is shown to achieve higher energy efficiency and shorter packet delays than predictable mobility.