This chapter utilizes a simple one-dimensional linear stability analysis which leads to a general approximate criterion from which one can derive the corresponding criterion for mudflows. It reviews the possible, theoretical approaches for the mathematical prediction of roll waves. The chapter describes a simple, general, theoretical analysis along with a physical interpretation of this phenomenon. It explores these results to mudflows considered as Herschel-Bulkley fluids. Roll waves have been frequently observed in mudflows in nature and in the laboratory. Roll waves in mudflows assume a special importance. Most works have concerned the stability criterion for simple fluid flows. The problem of the stability of the free surface flow of a yield stress fluid is basically the same as that of simple viscous fluids, since, as soon as it flows, the yield stress fluid has similar overall properties.