Neural implants have emerged over the last decade as highly effective solutions for the treatment of dysfunctions and disorders of the nervous system. These implants establish a direct, often bidirectional, interface to the nervous system, both sensing neural signals and providing therapeutic treatments. As a result of the technological progress and successful clinical demonstrations, completely implantable solutions have become a reality and are now commercially available for the treatment of various functional disorders. Central to this development is the wireless power transfer (WPT) that has enabled implantable medical devices (IMDs) to function for extended durations in mobile subjects. In this review, we present the theory, link design, and challenges, along with their probable solutions for the traditional near-field resonant inductively coupled WPT, capacitively coupled short-ranged WPT, and more recently developed ultrasonic, mid-field, and far-field coupled WPT technologies for implantable applications. A comparison of various power transfer methods based on their power budgets and WPT range follows. Power requirements of specific implants like cochlear, retinal, cortical, and peripheral are also considered and currently available IMD solutions are discussed. Patient's safety concerns with respect to electrical, biological, physical, electromagnetic interference, and cyber security from an implanted neurotech device are also explored in this review. Finally, we discuss and anticipate future developments that will enhance the capabilities of current-day wirelessly powered implants and make them more efficient and integrable with other electronic components in IMDs.