Intermetallic compounds exhibit a high degree of structural diversity, with numerous unique structure types. Among them, complex intermetallic compounds, also known as complex metallic alloys (CMAs), present large periodic structures based on building blocks consisting in highly symmetric atomic polyhedra, often displaying motifs of simpler compounds, like hierarchical replacement of atomic positions in the more elemental structures by larger atomic clusters or cages in the complex ones. Typical clusters/cages found in quasicrystals include Mackay, Bergman and Tsai clusters, and a large variety of highly symmetric polyhedra are generally found in CMAs. The structural complexity of CMAs extends to the surface. In this talk, the relationship between the three dimensional cluster-based bulk structure and the two-dimensional surface will be questioned in two cases: the pseudo-10fold surfaces of Al13TM4 (TM = Fe, Co) quasicrystalline approximants [1] and the low-index surfaces of the Ba8Au5.25Ge40.75 cage compounds [2], using a combination of experimental techniques under ultra-high vacuum and theoretical calculations based on the Density Functional Theory. The focus will be on the different mechanisms by which stable surface terminations are selected, to highlight the factors governing the atomic and electronic properties of the observed surface. The specific atomic arrangements at the surface induced by the intrinsic cluster-based bulk structure of CMAs are expected to affect their surface properties. This will be discussed in the case of the pseudo-10fold surfaces of Al13TM4 compounds, in terms of their catalytic properties [3]. [1] Ph. Scheid et al., Acta Cryst. A, 75 (2019) 314-324; É. Gaudry et al., Phys. Rev. B 94 (2016) 165406; J. Ledieu et al., Phys. Rev. Lett. 110 (2013) 076102. [2] K. Anand et al., J. Phys. Chem.. C 122 (2018) 2215 and 29298. [3] L. Piccolo et al., Sci. Technol. Adv. Mater. 20 (2019) 557.