We present a review of personality in neural conversational agents (CAs), also called chatbots. First, we define Personality, Persona, and Profile. We explain all personality schemes which have been used in CAs, and list models under the scheme(s) which they use. Second we describe 21 datasets which have been developed in recent CA personality research. Third, we define the methods used to embody personality in a CA, and review recent models using them. Fourth, we survey some relevant reviews on CAs, personality, and related topics. Finally, we draw conclusions and identify some research challenges for this important emerging field.