Wireless positioning technologies hold significant value for applications in autonomous driving, extended reality (XR), unmanned aerial vehicles (UAVs), and more. With the advancement of artificial intelligence (AI), leveraging AI to enhance positioning accuracy and robustness has emerged as a field full of potential. Driven by the requirements and functionalities defined in the 3rd Generation Partnership Project (3GPP) standards, AI/machine learning (ML)-based positioning is becoming a key technology to overcome the limitations of traditional methods. This paper begins with an introduction to the fundamentals of AI and wireless positioning, covering AI models, algorithms, positioning applications, emerging wireless technologies, and the basics of positioning techniques. Subsequently, focusing on standardization progress, we provide a comprehensive review of the evolution of 3GPP positioning standards, with an emphasis on the integration of AI/ML technologies in recent and upcoming releases. Based on the AI/ML-assisted positioning and direct AI/ML positioning schemes outlined in the standards, we conduct an in-depth investigation of related research. we focus on state-of-the-art (SOTA) research in AI-based line-of-sight (LOS)/non-line-of-sight (NLOS) detection, time of arrival (TOA)/time difference of arrival (TDOA) estimation, and angle estimation techniques. For Direct AI/ML Positioning, we explore SOTA advancements in fingerprint-based positioning, knowledge-assisted AI positioning, and channel charting-based positioning. Furthermore, we introduce publicly available datasets for wireless positioning and conclude by summarizing the challenges and opportunities of AI-driven wireless positioning.