The large models represented by ChatGPT have attracted a lot of attention from industry and academia for their excellent performance on text generation and semantic understanding tasks. The number of large model parameters has increased tens of thousands of times in three years and is still growing, which brings new challenges to storage systems. First, we analyze the storage challenges of large model training, pointing out that large model training has unique computation patterns, storage access patterns, and data characteristics, which makes traditional storage techniques inefficient in handling large model training tasks. Then, we describe three types of storage acceleration techniques and two types of fault-tolerant techniques. The storage acceleration techniques for large model training include: 1) distributed storage technique based on large model computation patterns, which designs the partitioning, storage, and transferring strategies of model data in distributed clusters based on the partitioning of large model computation tasks and the dependencies between computation tasks; 2) heterogeneous storage access pattern-aware technique for large model training, which develops data prefetching and transferring strategies among heterogeneous devices with the predictability of storage access patterns in large model training; 3) large model data reduction technique, which reduces the data size in the model training process according to the characteristics of large model data. The storage fault-tolerant techniques for large model training include: 1) parameter checkpointing technique, which stores the large model parameters to persistent storage devices; 2) redundant computation technique, which computes the same version of parameters repeatedly in multiple GPUs. Finally, we give the summary and suggestions for future research.