This paper presents the necessary and sufficient conditions for a third-order single-input/single-output linear system to have a nonovershooting (or monotone nondecreasing) step response. If the transfer function of an overall system has real poles, a necessary and sufficient condition is found for the nonovershooting (or monotone nondecreasing) step response. In the case of complex poles, one sufficient condition and two necessary conditions are obtained. The resulting conditions are all in terms of the coefficients of the numerator of the transfer function. Simple calculations can be used to check a system for the nonovershooting (or monotone nondecreasing) step response. Another feature is that the conditions in terms of pole-zero configurations can be easily derived from the present results.