To deal with the problem that multiple missiles cooperatively attack maneuvering targets with impact angle constraints, a novel leader-following missiles fixed-time cooperative guidance law was proposed. The leader-following missiles cooperative guidance model was established according to the relative motion between the target and missiles. Based on the fixed-time sliding mode control theory, the guidance law in the directions normal to the LOS(line-of-sight) was designed, so that the leader and followers could accurately attack the maneuvering target from the desired impact angles. Based on the consistency error function and the fixed-time sliding mode reaching law, the guidance law in the LOS direction of the followers was designed, which fully utilized the information exchange between missiles, and the time-to-go of the followers could quickly achieve consensus with that of the leaders in fixed time. Simulations and comparison verification indicate that the designed guidance law has higher control accuracy for time and angle constraints, no singularity and chattering phenomenon, faster convergence speed and the convergence time bound is independent of the initial states, which fully verifies the effectiveness and superiority of the proposed guidance law.