Centrosomes are composed of two orthogonally arranged centrioles surrounded by an electron-dense matrix called the pericentriolar material (PCM). Centrioles are cylinders with diameters of ~250 nm, are several hundred nanometres in length and consist of 9-fold symmetrically arranged microtubules (MT). In dividing animal cells, centrosomes act as the principal MT-organising centres and they also organise actin, which tunes cytoplasmic MT nucleation. In some specialised cells, the centrosome acquires additional critical structures and converts into the base of a cilium with diverse functions including signalling and motility. These structures are found in most eukaryotes and are essential for development and homoeostasis at both cellular and organism levels. The ultrastructure of centrosomes and their derived organelles have been known for more than half a century. However, recent advances in a number of techniques have revealed the high-resolution structures (at Å-to-nm scale resolution) of centrioles and have begun to uncover the molecular principles underlying their properties, including: protein components; structural elements; and biogenesis in various model organisms. This review covers advances in our understanding of the features and processes that are critical for the biogenesis of the evolutionarily conserved structures of the centrosomes. Furthermore, it discusses how variations of these aspects can generate diversity in centrosome structure and function among different species and even between cell types within a multicellular organism.