Wireless information and energy transfer (WIET) represents an emerging paradigm which employs controllable transmission of radio-frequency signals for the dual purpose of data communication and wireless charging. As such, WIET is widely regarded as an enabler of envisioned 6G use cases that rely on energy-sustainable Internet-of-Things (IoT) networks, such as smart cities and smart grids. Meeting the quality-of-service demands of WIET, in terms of both data transfer and power delivery, requires effective co-design of the information and energy signals. In this article, we present the main principles and design aspects of WIET, focusing on its integration in 6G networks. First, we discuss how conventional communication notions such as resource allocation and waveform design need to be revisited in the context of WIET. Next, we consider various candidate 6G technologies that can boost WIET efficiency, namely, holographic multiple-input multiple-output, near-field beamforming, terahertz communication, intelligent reflecting surfaces (IRSs), and reconfigurable (fluid) antenna arrays. We introduce respective WIET design methods, analyze the promising performance gains of these WIET systems, and discuss challenges, open issues, and future research directions. Finally, a near-field energy beamforming scheme and a power-based IRS beamforming algorithm are experimentally validated using a wireless energy transfer testbed. The vision of WIET in communication systems has been gaining momentum in recent years, with constant progress with respect to theoretical but also practical aspects. The comprehensive overview of the state of the art of WIET presented in this paper highlights the potentials of WIET systems as well as their overall benefits in 6G networks.