In this study, we present a unifying framework for the cones of tangents to an arbitrary set and some of its applications. We highlight the significance of these cones and their polars both from the point of view of differentiability and subdifferentiability theory and the point of view of mathematical programming. This leads to a generalized definition of a subgradient which extends the well-known definition of a subgradient of a convex function to the nonconvex case. As an application, we develop necessary optimality conditions for a min-max problem and show that these conditions are also sufficient under moderate convexity assumptions.