In spite of the considerable development in the last two decades of the state space approach to stability theory for linear time invariant systems the corresponding status of time varying and nonlinear systems is comparatively retarded. This apparent lack of maturity in the theory of variable coefficient and nonlinear differential equations can be ascribed to the need to derive the solutions of such systems before the structural properties of stability, controllability and observability can be ascertained. However for line-'ar time invariant systems such properties can be determined directly (or indirectly through the algebraic approach of Laplace transforms) in terms of the coefficient matrices. It is the prime purpose of this book to identify classes of linear and nonlinear multivariable time varying coefficient differential systems whose stability can be characterised directly from their variable coefficient matrices by a suitable transformation, in much the same manner as linear time invariant systems. A secondary purpose of this book is to collect together and unify recent advances in linear stability theory and to highlight those results which are directly applicable to practical dynamic systems.