The design of a fast iterative learning controller for a class of linear time-varying continuous systems with multiple time delays is discussed. A feedback iterative learning control (ILC) scheme is considered from the point of view of the two-dimensional (2D) system. To this end, a time-varying 2D error system is developed in the form of the continuous-discrete Roesser model, and its solution is summarised in terms of norm inequality. Based on this 2D error system, a sufficient convergence criterion is provided to ensure that the feedback ILC scheme drives the output tracking error to zero after just one iteration. Furthermore, by exploiting some prediction results with an observer, a new ILC scheme is derived only using feedforward controllers, which can equivalently achieve the learning performance of the feedback ILC scheme. The simulation results show that the proposed ILC schemes can provide fast updating laws to improve the tracking performance of the control systems.