A notable universal feature of wall-bounded turbulent flows is the universal logarithmic law of the wall deduced by Theodore von Kármán. This law of the wall describes how time-averaged streamwise velocity changes with distance from the wall. Despite the law of the wall having a universal von Kármán constant κ = 0.41 that governs the slope of the log-law velocity profile, as commonly known over a period of about 80 years, in fluvial streams there are a number of instances of the non-universality of κ. To be specific, it behaves as a variable in flows with low relative submergence, or where there is bed-load and/or suspended-load sediment transport. This article focuses on the aspect of non-universality of κ by inviting various open questions relating to future research directions.© Springer-Verlag Berlin Heidelberg 2013.