Autonomous underwater vehicles (AUVs) are increasingly used for undersea exploration. The endurance of AUVs is limited by the onboard energy storage among which the battery systems dominate. Various underwater recharging methods are employed to increase the AUV range and autonomy. Currently, contact-based underwater recharging utilizes the wet-mate connector technology that requires a high-precision AUV docking, and is prone to electrical safety issues. To overcome these limitations, underwater wireless recharging techniques for AUVs have been explored in recent years. Wireless charging offers a safe and reliable method for autonomous power transfer between a charging station and a vehicle. This article reviews the state-of-the-art inductive wireless power transfer (IWPT) solutions for underwater applications and discusses the engineering challenges of the IWPT system design. Underwater environmental factors, such as seawater conductivity, temperature, pressure, water currents, and biofouling phenomenon, impose constraints on IWPT systems. A comprehensive review of AUV energy storage systems, docking methods, IWPT system control methods, and compensation networks is presented in this article. Based on the main operational and constructional principles, the AUV IWPT systems are categorized as loosely coupled transformers and resonant IWPT systems. Each of the categories is illustrated through their main design principles and implementations reported in the literature so far. Technical challenges, such as integration of IWPT system into an AUV hull, interoperability, alignment and retention issues, docking station sinking and stability, the design of pressure-tolerant charging electronics, data transfer, and the battery operation in the underwater environment are discussed in this article too. The article is concluded with the best practice overview of designing an IWPT system for AUVs.