The study of consensus and controllability of multi-agent systems over finite fields is valuable to the control and application of various complex systems, including sensor networks and robot networks. Firstly, this survey investigates the consensus of multi-agent systems over finite fields, and algebraic criterion and graph theory criterion for achieving consensus are compared from the perspective of time delay and topological structure change. Then, the controllability of multi-agent systems over finite fields is considered. Five factors affecting controllability are introduced, including system model, topological structure, leader selection, domain order and controllability subscript. Finally, some potential future works are suggested.