We have averaged bright BATSE bursts to uncover the average overall temporal and spectral evolution of gamma-ray bursts (GRBs). We align the temporal structure of each burst by setting its duration to a standard duration, which we call T_. The observed average ``aligned $T_'' profile for 32 bright bursts with intermediate durations (16 to 40 s) has a sharp rise (within the first 20% of T_) and then a linear decay. Exponentials and power laws do not fit this decay. In particular, the power law seen in the x-ray afterglow (T^{-1.4}) is not observed during the bursts, implying that the x-ray afterglow is not just an extension of the x-ray evolution seen during the gamma-ray phase. The average burst spectrum has a low energy slope of -1.03, a high energy slope of -3.31, and a peak in the \nu F_{\nu} distribution at 390 keV. We determine the average spectral evolution. Remarkably, it is also a linear function, with the peak of the \nu F_{\nu} distribution given by 680 - 600 (T/T_) keV. Since both the temporal profile and the peak energy are linear functions, the peak energy is linearly proportional to the intensity. This behavior is inconsistent with the external shock model. Previously, trends have been reported for GRB evolution, but our results are quantitative relationships that models should attempt to explain.