The reason for the survival of mountainous topography in ancient orogenic belts is a long-standing problem in geomorphology and geodynamics. We explore the geomorphologic controls on the timescale for the postorogenic decay of topography to address the question of whether there is a viable geomorphologic explanation for the persistence of topography in ancient orogenic belts or whether alternative geodynamic explanations must be sought. Using both approximate analytical solutions and numerical simulations, we show that the standard detachment-limited stream power river incision model predicts, for reasonable initial topographies, relatively short postorogenic decay times of 1--10 Myr. Additional layers of complexity are introduced to this simplest model including isostasy and flexural strength, a transition to transport-limited conditions during decline, and the incorporation of the combined effects of a critical threshold for erosion and the stochastic variability of flood magnitudes. Each of these additional factors acts to lengthen the decay timescale. Pure Airy isostatic rebound of a thick crustal root results in decay times that are at most a factor of 6 longer than the detachment-limited result. The transition to transport-limited conditions involves partial protection of the bed by a thin layer of alluvium, which inhibits erosion and therefore increases decay times by a factor of 2--3 in our analysis. Finally, including critical shear stress results increases decay time by approximately a factor of 20 for the parameter values in our example calculation. More importantly, however, a significant portion of topography remains after predicted lowering rates have dropped to values less than measured rates of denudation in weathering-limited, low-relief environments. Thus a model combining isostasy, a transition to transport-limited conditions, and a critical shear stress for erosion could account for the presence of residual topography for hundreds of million years. All three of these factors can be expected to play a role in natural settings.