Automatic Question Answering (QA) has been successfully applied in various domains such as search engines and chatbots. Biomedical QA (BQA), as an emerging QA task, enables innovative applications to effectively perceive, access and understand complex biomedical knowledge. There have been tremendous developments of BQA in the past two decades, which we classify into 5 distinctive approaches: classic, information retrieval, machine reading comprehension, knowledge base and question entailment approaches. In this survey, we introduce available datasets and representative methods of each BQA approach in detail. Despite the developments, BQA systems are still immature and rarely used in real-life settings. We identify and characterize several key challenges in BQA that might lead to this issue, and discuss some potential future directions to explore.