Maintenance of an intact genome is essential for cellular and organismal homeostasis. The centromere is a specialized chromosomal locus required for faithful genome inheritance at each round of cell division. Human centromeres are composed of large tandem arrays of repetitive alpha-satellite DNA, which are often sites of aberrant rearrangements that may lead to chromosome fusions and genetic abnormalities. While the centromere has an essential role in chromosome segregation during mitosis, the long and repetitive nature of the highly identical repeats has greatly hindered in-depth genetic studies, and complete annotation of all human centromeres is still lacking. Here, we review our current understanding of human centromere genetics and epigenetics as well as recent investigations into the role of centromere DNA in disease, with a special focus on cancer, aging, and human immunodeficiency–centromeric instability–facial anomalies (ICF) syndrome. We also highlight the causes and consequences of genomic instability at these large repetitive arrays and describe the possible sources of centromere fragility. The novel connection between alpha-satellite DNA instability and human pathological conditions emphasizes the importance of obtaining a truly complete human genome assembly and accelerating our understanding of centromere repeats’ role in physiology and beyond.