Debugging is a vital aspect of software development, yet the debugging capabilities of Large Language Models (LLMs) remain largely unexplored. This paper first introduces DEBUGEVAL, a comprehensive benchmark designed to evaluate the debugging capabilities of LLMs. DEBUGEVAL collects data from existing high-quality datasets and designs four different tasks to evaluate the debugging effectiveness, including BUG Localization, BUG Identification, Code Review, and Code Repair. Additionally, to enhance the code debugging ability of LLMs, this paper proposes a CoMmunicative Agent BaSed DaTa REfinement FRamework (MASTER), which generates the refined code debugging data for supervised finetuning. Specifically, MASTER employs the Code Quizzer to generate refined data according to the defined tasks of DEBUGEVAL. Then the Code Learner acts as a critic and reserves the generated problems that it can not solve. Finally, the Code Teacher provides a detailed Chain-of-Thought based solution to deal with the generated problem. We collect the synthesized data and finetune the Code Learner to enhance the debugging ability and conduct the NeuDebugger model. Our experiments evaluate various LLMs and NeuDebugger in the zero-shot setting on DEBUGEVAL. Experimental results demonstrate that these 7B-scale LLMs have weaker debugging capabilities, even these code-oriented LLMs. On the contrary, these larger models (over 70B) show convincing debugging ability. Our further analyses illustrate that MASTER is an effective method to enhance the code debugging ability by synthesizing data for Supervised Fine-Tuning (SFT) LLMs.