  In this article we give a brief overview of some known results in the theory
of obstacle-type problems associated with a class of fourth-order elliptic
operators, and we highlight our recent work with collaborators in this
direction. Obstacle-type problems governed by operators of fourth order
naturally arise in the linearized Kirchhoff-Love theory for plate bending
phenomena. Moreover, as first observed by Yang in \cite{Y13}, boundary
obstacle-type problems associated with the weighted bi-Laplace operator can be
seen as extension problems, in the spirit of the one introduced by
Caffarelli-Silvestre, for the fractional Laplacian $(-\Delta)^s$ in the case
$1<s<2$. In our recent work, we investigate some problems of this type, where
we are concerned with the well-posedness of the problem, the regularity of
solutions, and the structure of the free boundary. In our approach, we combine
classical techniques from potential theory and the calculus of variations with
more modern methods, such as the localization of the operator and monotonicity
formulas.
