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 cellular positioning is becoming a key technology to overcome the limitations of traditional methods. This paper presents a comprehensive survey of AI-driven cellular positioning. We begin by reviewing the fundamentals of wireless positioning and AI models, analyzing their respective challenges and synergies. We provide a comprehensive review of the evolution of 3GPP positioning standards, with a focus on the integration of AI/ML in current and upcoming standard releases. Guided by the 3GPP-defined taxonomy, we categorize and summarize state-of-the-art (SOTA) research into two major classes: AI/ML-assisted positioning and direct AI/ML-based positioning. The former includes line-of-sight (LOS)/non-line-of-sight (NLOS) detection, time of arrival (TOA)/time difference of arrival (TDOA) estimation, and angle prediction; the latter encompasses fingerprinting, knowledge-assisted learning, and channel charting. Furthermore, we review representative public datasets and conduct performance evaluations of AI-based positioning algorithms using these datasets. Finally, we conclude by summarizing the challenges and opportunities of AI-driven wireless positioning.