AC, the Axiom of Choice, is the most controversial mathematical axiom due to its non-constructive nature, being avoided by some and used indiscriminately by others. This paper demonstrates paradigmatically that: - Catastrophes occur without AC: Many fundamental mathematical results fail (they are equivalent in ZF to AC or to a weak form of AC). - Catastrophes occur with AC: Many undesirable mathematical monsters are created (e.g., non-measurable sets and indeterminate games). - Some beautiful mathematical theorems hold only when AC is replaced by an alternative axiom that contradicts AC (e.g., by AD, the Axiom of Determinacy). Enlightening examples come from various areas of mathematics, particularly from general topology, but also from algebra, order theory, elementary analysis, measure theory, game theory, and graph theory.