So far, various aspects associated with wireless energy harvesting (EH) have been investigated from diverse perspectives, including energy sources and models, usage protocols, energy scheduling and optimization, and EH implementation in different wireless communication systems. However, a comprehensive survey specifically focusing on models of radio frequency (RF)-based EH behaviors has not yet been presented. To address this gap, this article provides an overview of the mainstream mathematical models that capture the nonlinear behavior of practical EH circuits, serving as a valuable handbook of mathematical models for EH application research. Moreover, we summarize the application of each nonlinear EH model, including the associated challenges and precautions. We also analyze the impact and advancements of each EH model on RF-based EH systems in wireless communication, utilizing artificial intelligence (AI) techniques. Additionally, we highlight emerging research directions in the context of nonlinear RF-based EH. This article aims to contribute to the future application of RF-based EH in novel communication research domains to a significant extent.