This paper studies the prescribed-time control problem of nonholonomic systems. By using some properties of a class of parametric Lyapunov equations and constructing time-varying Lyapunov-like functions, a time-varying high-gain feedback controller is firstly proposed for the chained nonholonomic system such that the state of the closed-loop system converges to zero at any prescribed time. The advantage of time-varying feedback is revealed by comparison with time-invariant feedback. Then time-varying high-gain feedback controllers for a class of perturbed nonholonomic systems are designed to achieve prescribed time convergence of the states. Both state feedback and observer-based output feedback are considered and the controls are proved to be bounded in all the cases. The effectiveness of the proposed methods are verified by numerical examples.