Knowledge base question answering (KBQA) aims to answer a question over a knowledge base (KB). Early studies mainly focused on answering simple questions over KBs and achieved great success. However, their performances on complex questions are still far from satisfactory. Therefore, in recent years, researchers propose a large number of novel methods, which looked into the challenges of answering complex questions. In this survey, we review recent advances in KBQA with the focus on solving complex questions, which usually contain multiple subjects, express compound relations, or involve numerical operations. In detail, we begin with introducing the complex KBQA task and relevant background. Then, we present two mainstream categories of methods for complex KBQA, namely semantic parsing-based (SP-based) methods and information retrieval-based (IR-based) methods. Specifically, we illustrate their procedures with flow designs and discuss their difference and similarity. Next, we summarize the challenges that these two categories of methods encounter when answering complex questions, and explicate advanced solutions as well as techniques used in existing work. After that, we discuss the potential impact of pre-trained language models (PLMs) on complex KBQA. To help readers catch up with SOTA methods, we also provide a comprehensive evaluation and resource about complex KBQA task. Finally, we conclude and discuss several promising directions related to complex KBQA for future research.