Substantial collective flow is observed in collisions between large nuclei at BNL RHIC (Relativistic Heavy Ion Collider) as evidenced by single-particle transverse momentum distributions and by azimuthal correlations among the produced particles. The data are well reproduced by perfect fluid dynamics. A calculation of the dimensionless ratio of shear viscosity eta to entropy density s by Kovtun, Son, and Starinets within anti-de Sitter space/conformal field theory yields eta/s=variant Planck's over 2pi/4pikB, which has been conjectured to be a lower bound for any physical system. Motivated by these results, we show that the transition from hadrons to quarks and gluons has behavior similar to helium, nitrogen, and water at and near their phase transitions in the ratio eta/s. We suggest that experimental measurements can pinpoint the location of this transition or rapid crossover in QCD.