The auditing of Large Language Models (LLMs) is a crucial and challenging task. In this study, we focus on auditing black-box LLMs without access to their parameters, only to the provided service. We treat this type of auditing as a black-box optimization problem where the goal is to automatically uncover input-output pairs of the target LLMs that exhibit illegal, immoral, or unsafe behaviors. For example, we may seek a non-toxic input that the target LLM responds to with a toxic output or an input that induces the hallucinative response from the target LLM containing politically sensitive individuals. This black-box optimization is challenging due to the scarcity of feasible points, the discrete nature of the prompt space, and the large search space. To address these challenges, we propose Curiosity-Driven Auditing for Large Language Models (CALM), which uses intrinsically motivated reinforcement learning to finetune an LLM as the auditor agent to uncover potential harmful and biased input-output pairs. CALM successfully identifies derogatory completions involving celebrities and uncovers inputs that elicit specific names under the black-box setting. This work offers a promising direction for auditing black-box LLMs. Content Warning: Please note that this paper includes examples that may be offensive.