LLM-based agent systems are emerging as a new software paradigm and have been widely adopted across diverse domains such as medicine, robotics, and programming. However, maintaining these systems requires substantial effort, as they are inevitably prone to bugs and continually evolve to meet changing external requirements. Therefore, automatically resolving agent issues (i.e.,bug reports or feature requests) is a crucial and challenging task. While recent software engineering (SE) agents (e.g., SWE-agent) have shown promise in addressing issues in traditional software systems, it remains unclear how effectively they can resolve real-world issues in agent systems, which differ significantly from traditional software. To fill this gap, we first manually analyze 201 real-world agent issues and identify common categories of agent issues. We then spend 500 person-hours constructing AgentIssue-bench, a reproducible benchmark comprising 50 agent issue resolution tasks (each with an executable environment and failure-triggering tests). We further evaluate state-of-the-art SE agents on AgentIssue-bench and reveal their limited effectiveness (\.e., with only 0.67% - 4.67% resolution rates). These results underscore the unique challenges of maintaining agent systems compared to traditional software, highlighting the need for further research to develop advanced SE agents for resolving agent issues.