  The paper is mainly devoted to systematic developments and applications of
geometric aspects of second-order variational analysis that are revolved around
the concept of parabolic regularity of sets. This concept has been known in
variational analysis for more than two decades while being largely
underinvestigated. We discover here that parabolic regularity is the key to
derive new calculus rules and computation formulas for major second-order
generalized differential constructions of variational analysis in connection
with some properties of sets that go back to classical differential geometry
and geometric measure theory. The established results of second-order
variational analysis and generalized differentiation, being married to the
developed calculus of parabolic regularity, allow us to obtain novel
applications to both qualitative and quantitative/numerical aspects of
constrained optimization including second-order optimality conditions,
augmented Lagrangians, etc. under weak constraint qualifications.
