We study the computational complexity of reasoning with global constraints. We show that reasoning with such constraints is intractable in general. We then demonstrate how the same tools of computational com- plexity can be used in the design and analysis of spe- cific global constraints. In particular, we illustrate how computational complexity can be used to determine when a lesser level of local consistency should be en- forced, when decomposing constraints will lose prun- ing,andwhencombiningconstraintsistractable.We also show how the same tools can be used to study symmetry breaking, meta-constraints like the cardinal- ity constraint, and learning nogoods.