PbZr 1-x Ti x O 3 (PZT) and Pb(Mg 1/3 Nb 2/3 ) 1-x Ti x O 3 (PMN-xPT) are complex lead-oxide perovskites that display exceptional piezoelectric properties for pseudorhombohedral compositions near a tetragonal phase boundary. In PZT these compositions are ferroelectrics, but in PMN-xPT they are relaxors because the dielectric permittivity is frequency dependent and exhibits non-Arrhenius behavior. We show that the nanoscale structure unique to PMN-xPT and other lead-oxide perovskite relaxors is absent in PZT and correlates with a greater than 100% enhancement of the longitudinal piezoelectric coefficient in PMN-xPT relative to that in PZT. By comparing dielectric, structural, lattice dynamical, and piezoelectric measurements on PZT and PMN-xPT, two nearly identical compounds that represent weak and strong random electric field limits, we show that quenched (static) random fields establish the relaxor phase and identify the order parameter.lead zirconate titanate | piezoelectricity | short-range order | soft modes | neutron scattering T he remarkable electromechanical properties of lead-oxide perovskite (ABO 3 ) relaxors such as Pb(Mg 1/3 Nb 2/3 ) 1-x Ti x O 3 (PMN-xPT) and Pb(Zn 1/3 Nb 2/3 ) 1-x Ti x O 3 (PZN-xPT) have inspired numerous attempts to understand the piezoelectricity in terms of the structural phase diagram, which contains a steep morphotropic phase boundary (MPB) separating pseudorhombohedral and tetragonal states over a narrow compositional range where the piezoelectricity is maximal (1-5). These materials exhibit very low strain-electric field hysteresis, extremely large dielectric constants, and record-setting piezoelectric coefficients at room temperature that form an unusually appealing set of properties with the potential to revolutionize a myriad of important technological applications spanning medical diagnostic sonography, military sonar, energy harvesting, and high-precision actuators (6-8). Many researchers have argued that quenched random electric fields (REFs) play a central role in establishing the relaxor phase, in part because the B sites of all known leadoxide perovskite relaxors are occupied by random mixtures of heterovalent cations (9-13). However, there is ample theoretical work that suggests relaxor behavior can occur in the absence of REFs (14-16). In fact it has not been proven conclusively that REFs are essential to the relaxor state or that they play any role in the ultrahigh piezoelectricity. These basic questions persist in the face of decades of research mainly because there exists no rigorous definition of what a relaxor is, i.e., there is no precise mathematical formulation of the relaxor-order parameter. To date, any material for which the real part of the dielectric permittivity e′ðω; TÞ exhibits a broad peak at a temperature T max that depends strongly (and in some cases only weakly) on the measuring frequency, ω, is classified as a relaxor. This definition has been applied equally to PMN and PZN, which possess strong REFs, as well as to specific compositions of K(T...