This paper addresses the leader-following consensus problem in multi-agent systems with time-varying coupling delays. The authors consider two different cases of coupling topologies: fixed and directed, and switched and balanced. For the fixed and directed topology, a necessary and sufficient condition for consensus is derived. For the switched and balanced topology, a sufficient condition is proposed. The analysis is based on Lyapunov-Razumikhin functions and linear matrix inequalities. Numerical examples are provided to illustrate the theoretical results. The paper also discusses the properties of the Laplacian matrix and the leader adjacency matrix, and provides conditions for the stability of the system. The findings are verified through simulations, showing that the proposed methods can effectively achieve consensus in multi-agent systems with time-varying coupling delays.This paper addresses the leader-following consensus problem in multi-agent systems with time-varying coupling delays. The authors consider two different cases of coupling topologies: fixed and directed, and switched and balanced. For the fixed and directed topology, a necessary and sufficient condition for consensus is derived. For the switched and balanced topology, a sufficient condition is proposed. The analysis is based on Lyapunov-Razumikhin functions and linear matrix inequalities. Numerical examples are provided to illustrate the theoretical results. The paper also discusses the properties of the Laplacian matrix and the leader adjacency matrix, and provides conditions for the stability of the system. The findings are verified through simulations, showing that the proposed methods can effectively achieve consensus in multi-agent systems with time-varying coupling delays.