The talk will focus on adaptive isogeometric methods for fourth-order problems employing almost-$C^1$ hierarchical spaces defined on multipatch geometries. Essentially, the idea is combining the construction presented in [3] with the hierarchical framework, in order to allow local refinement also in the solution of fourth-order problem, as an alternative to the already available options, see, e.g., [1,2,4]. It will be shown how this goal can be achieved by a careful analysis of the theoretical aspects, including the only partial nestedness of the biquadratic almost-$C^1$ spaces on the different levels of the hierarchy and the linear independence of their basis. We will highlight the features of the resulting isogeometic methods in a selection of examples. Bibliography [1] C. Bracco, C. Giannelli, M. Kapl, R. V\'azquez. Adaptive isogeometric methods with $C^1$ (truncated) hierarchical splines on planar multi-patch domains, Math. Mod. Meth. Appl. S. 33 (2023), 1829-1874. [2] C. Bracco, A. Farahat, C. Giannelli, M. Kapl, R. Vázquez, Adaptive methods with C1 splines for multi-patch surfaces and shells, Comput. Methods Appl. Mech. Engrg. 431 (2024). [3] T. Takacs, D. Toshniwal, Almost-$C^1$ splines: Biquadratic splines on unstructured quadrilateral meshes and their application to fourth order problems, Comput. Methods Appl. Mech. Engrg. 403 (2023). [4] P. Weinm\"uller, T. Takacs, An approximate $C^1$ multi-patch space for isogeometric analysis with a comparison to Nitsche's method, Comput. Methods Appl. Mech. Engrg. 401 (2022).