This paper focuses on energy-efficient packet transmission with individual packet delay constraints. The solution presented herein is a generalization of Uysal-Biyikoglu et al. (2002), which considered energy-efficient transmissions for a group of M packets subject to a single transmission deadline. First, the optimal offline scheduler (vis-a-vis total transmission energy) for packet transmissions with individual packet delay constraints is developed. It is shown that when packet inter-arrival times are independent and identically distributed (i.i.d.), the optimal transmission durations of packet m and packet M - m + 1, m isin 1,...,M, M ges 1, are identically distributed. This symmetry property leads to a simple and exact solution of the average packet delay for any i.i.d. inter-arrival times under the optimal offline scheduling. In addition, the packet delay performance for the single transmission deadline model is analyzed and shown to grow monotonically with M and at a rate proportional to radicM. A heuristic online scheduler, which assumes no future arrival information, is also studied and shown to achieve a comparable energy performance to the optimal offline scheduler in a wide range of scenarios. The flexible energy and delay tradeoff provided by the individual delay constraint model is further illustrated via simulations.