Achieving consistent high-quality machine translation (MT) across diverse domains remains a significant challenge, primarily due to the limited and imbalanced parallel training data available in various domains. While large language models (LLMs) have demonstrated impressive general understanding and generation abilities, their potential in multi-domain MT is under-explored. We establish a comprehensive benchmark for multi-domain translation, featuring 25 German$\Leftrightarrow$English and 22 Chinese$\Leftrightarrow$English test sets respectively covering 15 domains. Our evaluation of prominent LLMs reveals a discernible performance gap against traditional MT systems, highlighting domain overfitting and catastrophic forgetting issues after fine-tuning on domain-limited corpora. To mitigate this, we propose a domain Chain of Thought (CoT) fine-tuning technique that utilizes the intrinsic multi-domain intelligence of LLMs to improve translation performance. This method inspires the LLM to perceive domain information from the source text, which then serves as a helpful hint to guide the translation process. Despite being trained on a small dataset of four domains, our CoT fine-tune approach achieves notable enhancements in translation accuracy and domain robustness than traditional fine-tuning, as evidenced by an average 1.53 BLEU score increase in over 20 German$\rightarrow$English distinct out-of-domain tests.