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The broad spectrum available at millimeter-wave (mmWave) bands has attracted significant interest for a breadth of applications, with 5G communications being the main commercial drive for mmWave networks. Wireless power transmission and harvesting at mmWave bands have attracted significant attention due to the potential for minimizing the harvesting antenna size, allowing for more compact rectennas. For a fixed antenna size, the received power increases with frequency. Nevertheless, several challenges lie in realizing high efficiency antennas and rectifiers at mmWave bands. This article reviews the recent advances in mmWave rectenna design at a component- and system-level. Low-cost antennas and components for mmWave power harvesting, such as high efficiency scalable rectifiers on polymers and high radiation efficiency antennas on textiles, are reviewed. Both the antenna and rectifier can be realized using low-cost fabrication methods such as additively-manufactured circuits and packages, in addition to digital integrated circuits (ICs) for the rectifiers. Finally, this article provides an overview of future antenna design challenges and research directions for mmWave power harvesting.