In this thesis the stability and the region of attraction of nonlinear dynamical systems' equilibrium points are considered. Methods from linear programming are combined with theorems from the Lyapunov theory of dynamical systems to develop numerical algorithms. These algorithms deliver non-trivial information about the stability-behaviour of an equilibrium of a continuous, autonomous, nonlinear system. Two linear programs, LP1 and LP2, are developed. LP1 depends on a simply connected open neighborhood N of the equilibrium at the origin and two constants, a and m. The construction of LP1 implies that if it does not possess a feasible solution, then the corresponding system is not a,m-exponentially stable on N. LP2 has the property that every feasible solution of the linear program defines a piecewise-affine (piecewise-linear) Lyapunov function or a Lyapunov-like function V for the system.