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The development of large language models (LLMs) relies on sophisticated parallel training techniques, involving prolonged training runs with thousands of workers. Checkpointing systems are essential for handling failures in large-scale training. However, existing checkpointing systems are almost offline solutions tailored to specific parallelisms or model architectures. They lack adaptability to diverse parallel strategies and fail to recognize that most model states can be excluded from checkpoints, missing optimization opportunities. In this paper, we present AdaCheck, an adaptive checkpointing system that achieves minimized checkpoint size by characterizing and exploiting state redundancy across various parallelisms, model architectures, and training iterations. We model the state redundancy induced by parallelisms and model architectures using the abstraction tensor redundancy, and propose an offline redundancy utilization method to create checkpoints with a reduced set of states. To fully identify tensor redundancy, we design an efficient redundancy detector, which employs a hash-based data consistency check method and a ring-based communication algorithm. Besides, we introduce a novel online redundancy utilization method, which further reduces checkpoint size by exploiting the state redundancy across training iterations. Experimental results demonstrate that AdaCheck is adaptable to various parallelisms, including irregular parallelisms generated by automatic planners, as well as diverse model architectures, encompassing both dense and sparse architectures. Compared with state-of-the-art checkpointing approaches, AdaCheck can reduce checkpoint size by 6.00–896×, increase the checkpointing frequency by 1.46–111×, and incur almost no overhead on training throughput for LLM training.