A matrix is called totally positive (TP) if all its minors are positive. Such matrices enjoy a beautiful and rich theory and have found applications in various fields including statistics, approximation theory, mathematical biology and more. One of their most important properties is that multiplying a vector by a TP matrix can only decrease the number of sign changes in the vector. This property is known as the variation diminishing property (VDP) of TP matrices. It has been recently shown that many strong and interesting results on the stability of nonlinear dynamical systems actually follow from the VDP. The course includes two parts. The first reviews totally positive matrices and their rich set of special properties. The second part describes their applications to the stability analysis of nonlinear dynamical systems.