Centromeric chromatin is distinguished primarily by nucleosomes containing the histone variant cenH3, which organizes the kinetochore that links the chromosome to the spindle apparatus. Whereas budding yeast have simple 'point' centromeres with single cenH3 nucleosomes, and fission yeast have 'regional' centromeres without obvious sequence specificity, the centromeres of most organisms are embedded in highly repetitive 'satellite' DNA. Recent studies have revealed a remarkable diversity in centromere chromatin organization among different lineages, including some that have lost cenH3 altogether. We review recent progress in understanding point, regional and satellite centromeres, as well as less well-studied centromere types, such as holocentromeres. We also discuss the formation of neocentromeres, the role of pericentric heterochromatin, and the structure and composition of the cenH3 nucleosome. 