Many optimization problems can be equivalently reduced to solving equations of the form \(F(x) = y\) , where F is a mapping that may not be Fréchet differentiable, thus preventing the direct use of the classical Newton method. In this chapter, we introduce the semi-smooth Newton method and demonstrate its superlinear convergence under the concept of Newton differentiability of F, a condition that is much weaker than Fréchet differentiability. Newton differentiability can be guaranteed by semi-smoothness, which is defined in terms of Clarke’s generalized Jacobian. It turns out that a large class of tame functions are semi-smooth. We also discuss a globalization of the semi-smooth Newton method and present various applications in optimization.