In this chapter, our focus will be on a differential object, namely the limiting subdifferential. Our aim here is to show the key role it plays in non-convex optimization. Along our way, we shall also discuss some key subdifferential concepts, like proximal subdifferential or regular subdifferential, whose limiting version results in the object of our interest. Subdifferentials mimic the derivative at a point where it is absent. Thus, we will also discuss the calculus rules and how they are the key in developing necessary optimality conditions. Further, all these issues are deeply linked to the nonsmooth geometry of tangents and normals, which will also play a key role in our discussion. Our attempt would be to review everything with a fresh eye and further show how the proximal subdifferential often offers an easy approach to the key results of the limiting subdifferential. We will keep our discussions firmly in finite dimensions.