The Routh-Hurwitz criterion is a mathematical test used in control theory to determine the stability of a linear time-invariant system. It analyzes the coefficients of the characteristic polynomial of a system's transfer function to assess whether all roots have negative real parts, which indicates stability. By constructing a Routh array, the criterion provides a systematic way to check the sign changes in the first column of the array. If there are no sign changes, the system is stable. This method is particularly useful for engineers and researchers working with control systems and dynamical systems.