  Variational inequalities are an important mathematical tool for modelling
free boundary problems that arise in different application areas. Due to the
intricate nonsmooth structure of the resulting models, their analysis and
optimization is a difficult task that has drawn the attention of researchers
for several decades. In this paper we focus on a class of variational
inequalities, called of the second kind, with a twofold purpose. First, we aim
at giving a glance at some of the most prominent applications of these types of
variational inequalities in mechanics, and the related analytical and numerical
difficulties. Second, we consider optimal control problems constrained by these
variational inequalities and provide a thorough discussion on the existence of
Lagrange multipliers and the different types of optimality systems that can be
derived for the characterization of local minima. The article ends with a
discussion of the main challenges and future perspectives of this important
problem class.
