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Nonlinear systems have received little attention in conjunction with the problem of achieving a non-overshooting tracking response. This paper demonstrates a means of obtaining a non-overshooting tracking response for single- input/single-output strict-feedback nonlinear systems. With the proposed method, arbitrary reference trajectories can be tracked "from below" for arbitrary initial conditions (as long as the initial value of the plant output is strictly below the initial value of the reference trajectory). In addition, a design is presented for "approximately" non-overshooting control in the presence of disturbances, where the amount of overshoot can be made arbitrarily small by appropriately choosing the control gains. Finally, an output-feedback example shows the ability of our approach to ensure arbitrarily small overshoot, where the overshoot is caused by the initial condition of the unmeasured part of the state.