Chatbots have been studied for more than half a century. With the rapid development of natural language processing (NLP) technologies in recent years, chatbots using large language models (LLMs) have received much attention nowadays. Compared with traditional ones, modern chatbots are more powerful and have been used in real-world applications. There are however, bias and fairness concerns in modern chatbot design. Due to the huge amounts of training data, extremely large model sizes, and lack of interpretability, bias mitigation and fairness preservation of modern chatbots are challenging. Thus, a comprehensive overview on bias and fairness in chatbot systems is given in this paper. The history of chatbots and their categories are first reviewed. Then, bias sources and potential harms in applications are analyzed. Considerations in designing fair and unbiased chatbot systems are examined. Finally, future research directions are discussed.