The study of consensus and controllability of multi-agent systems in finite fields is valuable to the control and application of various complex systems, including sensor networks and robot networks. The concept of consensus of multi-agent systems refers to: there is a finite time moment at which the state values of all agents are equal to some values on a finite field. The controllability of a multi-agent system over finite fields is defined as: by applying appropriate control inputs to certain nodes, the system can be driven to any target state. Firstly, this paper gives a review of the consensus of multi-agent systems over finite fields. Then, we provide a comprehensive survey of the connections, distinctions of algebraic criterion and graph theory criterion for achieving consensus from the perspective of time delay and topological structure change. Secondly, the controllability of multi-agent systems over finite fields is presented, five factors affecting controllability are summarized, including system model, topological structure, leader selection, domain order and controllability subscript. Finally, some potential future works are suggested.