This paper develops nonsmooth Lyapunov stability theory and LaSalle's invariance principle for a class of nonsmooth Lipschitz continuous Lyapunov functions and absolutely continuous state trajectories. Computable tests based on Filipov's differential inclusion and Clarke's generalized gradient are derived. The primary use of these results is in analyzing the stability of equilibria of differential equations with discontinuous right-hand sides such as in nonsmooth dynamic systems or variable structure control. The paper introduces the concept of Filipov solutions for differential equations with discontinuous right-hand sides, and develops a connection via a new chain rule for differentiating regular functions along Filipov solution trajectories. Clarke's generalized gradient is also discussed, which is particularly useful in simplifying proofs in nonsmooth analysis. Two stability theorems are stated in terms of the set valued map $\dot{V}$. The first theorem establishes uniform stability and uniform asymptotic stability under certain conditions. The second theorem is a nonsmooth version of LaSalle's theorem, which states that every trajectory in a compact set converges to the largest invariant set in the closure of the set where $\dot{V} = 0$. The paper also provides an example of a nonsmooth dynamic system with Coulomb friction, and shows that the equilibrium point cannot be shown stable by any smooth time independent Lyapunov function. The example demonstrates the importance of nonsmooth Lyapunov functions in analyzing such systems. The paper concludes by stating that the developed theory is applicable to systems with switches, where natural Lyapunov functions are often only piecewise smooth. This machinery should find application in variable structure control theory, the analysis and control of mechanical systems, and the analysis of pulse width modulated control systems.This paper develops nonsmooth Lyapunov stability theory and LaSalle's invariance principle for a class of nonsmooth Lipschitz continuous Lyapunov functions and absolutely continuous state trajectories. Computable tests based on Filipov's differential inclusion and Clarke's generalized gradient are derived. The primary use of these results is in analyzing the stability of equilibria of differential equations with discontinuous right-hand sides such as in nonsmooth dynamic systems or variable structure control. The paper introduces the concept of Filipov solutions for differential equations with discontinuous right-hand sides, and develops a connection via a new chain rule for differentiating regular functions along Filipov solution trajectories. Clarke's generalized gradient is also discussed, which is particularly useful in simplifying proofs in nonsmooth analysis. Two stability theorems are stated in terms of the set valued map $\dot{V}$. The first theorem establishes uniform stability and uniform asymptotic stability under certain conditions. The second theorem is a nonsmooth version of LaSalle's theorem, which states that every trajectory in a compact set converges to the largest invariant set in the closure of the set where $\dot{V} = 0$. The paper also provides an example of a nonsmooth dynamic system with Coulomb friction, and shows that the equilibrium point cannot be shown stable by any smooth time independent Lyapunov function. The example demonstrates the importance of nonsmooth Lyapunov functions in analyzing such systems. The paper concludes by stating that the developed theory is applicable to systems with switches, where natural Lyapunov functions are often only piecewise smooth. This machinery should find application in variable structure control theory, the analysis and control of mechanical systems, and the analysis of pulse width modulated control systems.