Full-parameter fine-tuning is computationally prohibitive for large language models (LLMs), making parameter-efficient fine-tuning (PEFT) methods like low-rank adaptation (LoRA) increasingly popular. However, LoRA and its existing variants introduce significant latency in multi-tenant settings, hindering their applications in the industry. To address this issue, we propose the Fantastic LoRA (FanLoRA) framework, which consists of four steps: (a) adding LoRA modules to all the Transformer linear weights and fine-tuning on a large-scale instruction tuning dataset. (b) The importance of each module is then assessed using a novel importance scoring method. (c) only the most critical modules per layer are retained, resulting in the FanLoRA setting. (d) The FanLoRA setting is applied to fine-tune various downstream tasks. Our extensive experiments demonstrate that: (a) FanLoRA outperforms existing PEFT baselines across a wide collection of tasks with comparable tunable parameters. (b) FanLoRA significantly reduces the inference latency of LoRA, making it valuable for further broadening the applications of LLMs in the industry.