Nonsmoothness arises in PDE-constrained optimization in multi-faceted ways and is at the core of many recent developments in this field. This talk highlights selected situations where nonsmoothness occurs, either directly or by reformulation, and discusses how it can be handled analytically and numerically. Projections, proximal maps, and similar techniques provide powerful tools to convert variational inequalities and stationarity conditions into nonsmooth operator equations. These then enable the application of semismooth Newton methods, which belong to the most successful algorithms for inequality constrained problems with PDEs. Nonsmoothness also arises in problems with equilibrium constraints, such as MPECs, or with bilevel structure. This is particularly apparent when reduced problems are generated via solution operators of subsystems. Similarly, nonlinear PDEs, even if looking smooth at first sight, often pose challenges regarding the differentiability of the solution operator with respect to parameters such as control or shape. There are also important applications where nonsmoothness arises naturally in the cost function, for instance in sparse control or when risk measures like the CVaR are used in risk-averse PDE-constrained optimization. Also other approaches for PDE-constrained optimization under uncertainty exhibit nonsmoothness, e.g., (distributionally) robust optimization. All this shows that PDE-constrained optimization with nonsmooth structures is a broad and highly relevant field. This talk discusses specific examples to highlight the importance of being able to master nonsmoothness in PDE optimization. General patterns are identified and parallels to the finite dimensional case are explored. The infinite dimensional setting of PDEs, however, poses additional challenges and unveils problem characteristics that are still present after discretization but then are much less accessible to a rigorous analysis.