Abstract: We are dedicated to addressing the fully distributed prescribed-time time-varying formation tracking problem for linear multi-agent systems subject to uncertainties and disturbances. First, a fully distributed prescribed-time observer is designed for each follower agent to estimate the state of the leader. Its advantage lies in not only avoiding the use of global information such as eigenvalues of the Laplacian matrix and the number of agents in the observer design, but also enabling precise estimation of the leader's state within a prescribed-time. Second, based on the estimation of the leader's state, a fully distributed prescribed-time piecewise controller is designed for each follower. This controller incorporates a robust term and a time-varying formation compensation term, as well as a time-varying scaling function, effectively counteracting the negative effects caused by system uncertainties and external disturbances, thereby achieving the prescribed-time time-varying formation tracking. Finally, the effectiveness of the proposed controller is verified through a numerical simulation.