This paper presents a comprehensive survey of artificial intelligence (AI)-based cellular positioning technologies. It highlights the importance of wireless positioning technologies and the potential of AI utilization. It examines the development of AI/machine learning (ML)-based cellular positioning technologies based on the requirements and capabilities defined in the 3GPP standards. It analyzes the evolution of the 3GPP positioning standard and examines current and future standard versions, focusing on AI/ML integration. It categorizes and summarizes state-of-the-art (SOTA) research into two main categories: AI/ML-assisted positioning and direct AI/ML-based positioning. The former includes LOS/NLOS detection, TOA/TDOA estimation, and angle estimation, while the latter encompasses fingerprinting, knowledge-assisted learning, and channel charting. Representative public datasets are reviewed and the performance of AI-based positioning algorithms is evaluated using these datasets. Finally, the challenges and opportunities for AI-based wireless positioning are summarized.