This paper presents a simultaneous task assignment and trajectory planning method for unmanned system swarm by using optimal transport and model predictive control (OT-MPC). Unlike the conventional hierarchical assignment and planning, the proposed approach addresses both the task assignment and trajectory planning subproblems concurrently. To be specific, a unified cost function is designed to solve task assignment and trajectory planning problem. Moreover, the multi-tasks are assigned by using optimal transport, which establishes an optimal mapping between tasks and unmanned system vehicles based on transportation cost. The trajectory planning is achieved by using model predictive control, which generates high-quality navigation trajectories considering obstacle avoidance. Finally, the proposed method is applied to the unmanned surface vehicles swarm. Numerical simulations and experiments were conducted to validate the effectiveness of the proposed method.