In this talk, we discuss the role that subgradients play in various second-order variational analysis constructions and its consequences. Focusing mainly on the behavior of the second subderivative and subgradient proto-derivative of certain composite functions, we demonstrate that choosing the underlying subgradient, utilized in the definitions of these concepts, from the relative interior of the subdifferential mapping ensures stronger second-order variational properties such as strict twice epi-differentiability and strict subgradient proto-differentiability. Using this observation, we provide a simple characterization of continuous differentiability of the proximal mapping of our composite functions. As another application, we discuss the equivalence of metric regularity and strong metric regularity of a class of generalized equations at their nondegenerate solutions. This talk is based on joint works with Nguyen T. V. Hang. Speaker(s): Ebrahim Sarabi (Miami University)