The centriole and basal body (CBB) structure nucleates cilia and flagella, and is an essential component of the centrosome, underlying eukaryotic microtubule-based motility, cell division and polarity. In recent years, components of the CBB-assembly machinery have been identified, but little is known about their regulation and evolution. Given the diversity of cellular contexts encountered in eukaryotes, but the remarkable conservation of CBB morphology, we asked whether general mechanistic principles could explain CBB assembly. We analysed the distribution of each component of the human CBB-assembly machinery across eukaryotes as a strategy to generate testable hypotheses. We found an evolutionarily cohesive and ancestral module, which we term UNIMOD and is defined by three components (SAS6, SAS4/CPAP and BLD10/CEP135), that correlates with the occurrence of CBBs. Unexpectedly, other players (SAK/PLK4, SPD2/CEP192 and CP110) emerged in a taxon-specific manner. We report that gene duplication plays an important role in the evolution of CBB components and show that, in the case of BLD10/CEP135, this is a source of tissue specificity in CBB and flagella biogenesis. Moreover, we observe extreme protein divergence amongst CBB components and show experimentally that there is loss of cross-species complementation among SAK/PLK4 family members, suggesting species-specific adaptations in CBB assembly. We propose that the UNIMOD theory explains the conservation of CBB architecture and that taxon- and tissue-specific molecular innovations, gained through emergence, duplication and divergence, play important roles in coordinating CBB biogenesis and function in different cellular contexts.