Centromeres are chromosomal loci that are defined epigenetically in most eukaryotes by incorporation of a centromere-specific nucleosome in which the canonical histone H3 variant is replaced by Centromere Protein A (CENP-A). Therefore, the assembly and propagation of centromeric nucleosomes are critical for maintaining centromere identify and ensuring genomic stability. Centromeres direct chromosome segregation (during mitosis and meiosis) by recruiting the constitutive centromere-associated network of proteins throughout the cell cycle that in turn recruits the kinetochore during mitosis. Assembly of centromere-specific nucleosomes in humans requires the dedicated CENP-A chaperone HJURP, and the Mis18 complex to couple the deposition of new CENP-A to the site of the pre-existing centromere, which is essential for maintaining centromere identity. Human CENP-A deposition occurs specifically in early G1, into pre-existing chromatin, and several additional chromatin-associated complexes regulate CENP-A nucleosome deposition and stability. Here we review the current knowledge on how new CENP-A nucleosomes are assembled selectively at the existing centromere in different species and how this process is controlled to ensure stable epigenetic inheritance of the centromere.