Expansions of simple tandem repeats are responsible for almost 50 human diseases, the majority of which are severe, degenerative, and not currently treatable or preventable. In this review, we first describe the molecular mechanisms of repeat-induced toxicity, which is the connecting link between repeat expansions and pathology. We then survey alternative DNA structures that are formed by expandable repeats and review the evidence that formation of these structures is at the core of repeat instability. Next, we describe the consequences of the presence of long structure-forming repeats at the molecular level: somatic and intergenerational instability, fragility, and repeat-induced mutagenesis. We discuss the reasons for gender bias in intergenerational repeat instability and the tissue specificity of somatic repeat instability. We also review the known pathways in which DNA replication, transcription, DNA repair, and chromatin state interact and thereby promote repeat instability. We then discuss possible reasons for the persistence of disease-causing DNA repeats in the genome. We describe evidence suggesting that these repeats are a payoff for the advantages of having abundant simple-sequence repeats for eukaryotic genome function and evolvability. Finally, we discuss two unresolved fundamental questions: (i) why does repeat behavior differ between model systems and human pedigrees, and (ii) can we use current knowledge on repeat instability mechanisms to cure repeat expansion diseases?