## 1 Introduction  
Graphene, the two-dimensional (2D) sp\({}^{2}\) allotropic form of carbon, has been the object of continuously increasing scientific and technological interest, starting from its first isolation in 2004 [1]. Among the different types of graphene materials considered so far, the epitaxial graphene (EG) grown by controlled high temperature graphitization of the (0001) face of hexagonal silicon carbide (6H- or 4H-SiC) [2; 3; 4] is a material of choice for advanced applications in electronics [5; 6], high precision metrology [7] and environmental sensing [8]. The main advantage of this growth method over other commonly used approaches (such as chemical vapor deposition, CVD, on catalytic metals [9]) is the availability of high quality graphene directly on a semiconducting or semi-insulating substrate, i.e., ready for electronic devices fabrication, without the need of transfer processes [10; 11] typically responsible for contaminations and damages [12; 13]. Deposition of uniform ultra-thin insulators (especially high-k dielectrics) on graphene is a key step for the fabrication of graphene-based electronic devices [6; 14; 15; 16; 17]. In this context, the atomic layer deposition (ALD), owing to its sequential layer-by-layer growth mechanism [18], is the most suitable candidate to achieve a conformal growth with subnanometric control on the thickness. However, the lack of out-of-plane bonds or surface groups in the sp\({}^{2}\) lattice of graphene typically represents the main obstacle to films nucleation in the early stage of the ALD growth. This typically results in an inhomogeneous coverage of graphene surface, although the quality of the deposited films strongly depends on the graphene synthesis method, the graphene substrate and eventual transfer processes of graphene from the native substrate to foreign ones. As an example, for defect-free graphene flakes exfoliated from graphite, ALD growth was found to occur preferentially at the edges of the flakes [19], whereas, in the case of polycrystalline graphene grown by CVD on metals and transferred to insulating substrates, the ALD nucleation typically occurs at the grain boundaries and on wrinkles [20]. Interestingly, in the specific case of EG grown by thermal decomposition of SiC (0001), uniform ALD coverage has been observed on the monolayer (1 L) EG areas, whereas inhomogeneous growth has been found on the bilayer (2 L) of few layer regions [21]. These ALD nucleation issues common to all the graphene materials are typically circumvent by adopting surface preparation protocols, consisting of chemical prefunctionalization of graphene surface to introduce reactive (sp\({}^{3}\)) sites, or by the predeposition of seed layers [22; 23; 24; 25; 26; 27; 28; 29; 30; 31; 32]. However, proper optimization of these processes is necessary in order to avoid structural damages and a degradation of graphene electronic properties. Although the research on ALD for graphene devices integration has been recently the object of comprehensive review articles [33], to the best of our knowledge, a focused paper reviewing ALD of high-k insulators on the EG/SiC(0001) system is currently missing. This article aims to provide an overview on this topic, highlighting the specific approaches adopted to achieve uniform ALD growth in the case of EG. Section 2 provides an introduction to the peculiar structural and electrical properties of EG and its interface with the (0001) SiC substrate. Section 3 presents recent results on the direct deposition of Al\({}_{2}\)O\({}_{3}\) thin films on the pristine EG, elucidating the key role played by the sp\({}^{3}\) hybridized buffer layer to obtain a homogeneous Al\({}_{2}\)O\({}_{3}\) coverage on monolayer EG. Furthermore, the ALD on EG coated with different seeding layers, including oxidized metal films [34], high-k metal oxide thin films [35], self-assembled organic monolayers [36] or spin coated graphene oxide [37], will be presented in Section 4.