The stream power incision model (SPIM) is a cornerstone of quantitative geomorphology. It states that river incision rate is the product of drainage area and channel slope raised to the power exponents m and n, respectively. It is widely used to predict patterns of deformation from channel long profile inversion or to model knickpoint migration and landscape evolution. Numerous studies have attempted to test its applicability with mixed results prompting the question of its validity. This paper synthesizes these results, highlights the SPIM deficiencies, and offers new insights into the role of incision thresholds and channel width. By reviewing quantitative data on incising rivers, I first propose six sets of field evidence that any long-term incision model should be able to predict. This analysis highlights several inconsistencies of the standard SPIM. Next, I discuss the methods used to construct physics-based long-term incision laws. I demonstrate that all published incising river datasets away from knickpoints or knickzones are in a regime dominated by threshold effects requiring an explicit upscaling of flood stochasticity neglected in the standard SPIM and other incision models. Using threshold-stochastic simulations with dynamic width, I document the existence of composite transient dynamics where knickpoint propagation locally obeys a linear SPIM (n=1) while other part of the river obey a non-linear SPIM (n>1). The threshold-stochastic SPIM resolves some inconsistencies of the standard SPIM and matches steady-state field evidence when width is not sensitive to incision rate. However it fails to predict the scaling of slope with incision rate for cases where width decreases with incision rate. Recent proposed models of dynamic width cannot resolve these deficiencies. An explicit upscaling of sediment flux and threshold-stochastic effects combined with dynamic width should take us beyond the SPIM which is shown here to have a narrow range of validity. Copyright © 2013 John Wiley & Sons, Ltd.