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As wireless has disrupted communications, wireless will also disrupt the delivery of energy. Future wireless networks will be equipped with (radiative) wireless power transfer (WPT) capability and exploit radio waves to carry both energy and information through a unified wireless information and power transfer (WIPT). Such networks will make the best use of the RF spectrum and radiation as well as the network infrastructure for the dual purpose of communicating and energizing. Consequently those networks will enable trillions of future low-power devices to sense, compute, connect, and energize anywhere, anytime, and on the move. In this paper, we review the foundations of such future system. We first give an overview of the fundamental theoretical building blocks of WPT and WIPT. Then we discuss some state-of-the-art experimental setups and prototypes of both WPT and WIPT and contrast theoretical and experimental results. We draw a special attention to how the integration of RF, signal and system designs in WPT and WIPT leads to new theoretical and experimental design challenges for both microwave and communication engineers and highlight some promising solutions. Topics and experimental testbeds discussed include closed-loop WPT and WIPT architectures with beamforming, waveform, channel acquisition, and single/multi-antenna energy harvester, centralized and distributed WPT, reconfigurable metasurfaces and intelligent surfaces for WPT, transmitter and receiver architecture for WIPT, modulation, rate-energy trade-off. Moreover, we highlight important theoretical and experimental research directions to be addressed for WPT and WIPT to become a foundational technology of future wireless networks.