The transformative power of artificial intelligence (AI) and machine learning (ML) is recognized as a key enabler for sixth generation (6G) mobile networks by both academia and industry. Research on AI/ML in mobile networks has been ongoing for years, and the 3rd generation partnership project (3GPP) launched standardization efforts to integrate AI into mobile networks. However, a comprehensive review of the current status and challenges of the standardization of AI/ML for mobile networks is still missing. To this end, we provided a comprehensive review of the standardization efforts by 3GPP on AI/ML for mobile networks. This includes an overview of the general AI/ML framework, representative use cases (i.e., CSI feedback, beam management and positioning), and corresponding evaluation matrices. We emphasized the key research challenges on dataset preparation, generalization evaluation and baseline AI/ML models selection. Using CSI feedback as a case study, given the test dataset 2, we demonstrated that the pre-training-fine-tuning paradigm (i.e., pre-training using dataset 1 and fine-tuning using dataset 2) outperforms training on dataset 2. Moreover, we observed the highest performance enhancements in Transformer-based models through fine-tuning, showing its great generalization potential at large floating-point operations (FLOPs). Finally, we outlined future research directions for the application of AI/ML in mobile networks.