This article classifies, describes, and critically compares different compensation schemes, converter topologies, control methods, and coil structures of wireless power transfer systems for electric vehicle battery charging, focusing on inductive power transfer. It outlines a path from the conception of the technology to the modern and cutting edge of the technology. First, the base principles of inductive coupling power transfer are supplied to give an appreciation for the operation and design of the systems. Then, compensation topologies and soft-switching techniques are introduced. Reimagined converter layouts that deviate from the typical power electronics topologies are introduced. Control methods are detailed alongside topologies, and the generalities of control are also included. The paper then addresses other essential aspects of wireless power transfer systems such as coil design, infrastructure, cost, and safety standards to give a broader context for the technology. Discussions and recommendations are also provided. This paper aims to explain the technology, its modern advancements, and its importance. With the need for electrification mounting and the automotive industry being at the forefront of concern, recent advances in wireless power transfer will inevitably play an essential role in the coming years to propel electric vehicles into the common mode of choice.