The objective of this study is to systematically elucidate the transient rheological behavior of viscoelastic polymer solutions at start-up of steady shear flow. Using a strain-controlled rheometer [Advanced Rheometric Expansion System (ARES)], the stress overshoot behavior of concentrated aqueous poly(ethylene oxide) [PEO] solutions has been thoroughly investigated with applying a number of shear rates to these polymer solutions. In this article, the typical stress growth behavior of aqueous PEO solutions was firstly presented on the basis of experimentally obtained data and nextly the effects of shear rate, molecular weight, and concentration on this behavior were discussed in depth. In order to theoretically predict the stress growth behavior, the Wagner constitutive equation (a time-strain separable constitutive equation) and the Bird-Leider model were employed and the applicability of these models was examined in detail. The main findings obtained from this study can be summarized as follows: (1) At the inception of low shear rates, the stress overshoot behavior is not observed but the shear stress progressively increases with time until the steady state stress value is reached. As the shear rate is increased, however, aqueous PEO solutions exhibit a pronounced stress overshoot followed by a stress decay to reach the steady state flow condition. (2) The maximum shear stress increases linearly with an increase in shear rate in a double logarithmic scale and becomes larger with an increase in concentration. (3) The time at which the maximum shear stress occurs, $t_{max}$, shows a linear relationship with the inverse of shear rate in a double logarithmic scale, regardless of molecular weight and concentration. (4) When the Wagner constitutive equation is used, the Wagner damping function exhibits superior performance to the Soskey-Winter damping function in predicting the transient stress growth behavior of aqueous PEO solutions. (5) The Bird-Leider empirical model can be successfully used for predicting the transient stress growth behavior of aqueous PEO solutions. This model is especially effective for expressing the maximum shear stress as well as the time at which this maximum stress is observed.