Relaxor-ferroelectrics are fascinating and useful materials, but the mechanism of relaxor-ferroelectricity has been puzzling the scientific community for more than 65 years. Here, a theory of relaxorferroelectricity is presented based on 3-dimensional-extended-random-site-Ising-model along with Glauber-dynamics of pseudospins. We propose a new mean-field of pseudospin-strings to solve this kinetic model. The theoretical results show that, with decreasing pseudospin concentration, there are evolutions from normal-ferroelectrics to relaxor-ferroelectrics to paraelectrics, especially indicating by the crossovers from, (a) the sharp to diffuse change at the phase-transition temperature to disappearance in the whole temperature range of order-parameter, and (b) the power-law to Vogel-Fulcher-law to Arrheniusrelation of the average relaxation time. Particularly, the calculated local-order-parameter of the relaxorferroelectrics gives the polar-nano-regions appearing far above the diffuse-phase-transition and shows the quasi-fractal characteristic near and below the transition temperature. We also provide a new mechanism of Burns-transformation which stems from not only the polar-nano-regions but also the correlationfunction between pseudospins, and put forward a definition of the canonical relaxor-ferroelectrics. The theory accounts for the main facts of relaxor-ferroelectricity, and in addition gives a good quantitative agreement with the experimental results of the order-parameter, specific-heat, high-frequency permittivity, and Burns-transformation of lead magnesium niobate, the canonical relaxor-ferroelectric. 65 years after the discovery of so-called relaxor-ferroelectrics (RFEs) 1 , this manuscript promises to deliver the still missing theory of relaxor-ferroelectricity [Supplementary Information (SI) 1] 2-10 . For the existing phase-transition theories of normal-ferroelectrics are based on both structure and component homogeneity [11][12][13] , theoretically, the main difficulty in describing relaxor-ferroelectricity originates from RFEs being component-disordered although structure-ordered, i.e. disordered components on crystal lattices [14][15][16][17][18][19] . In fact, understanding how the component disorder on lattices leads to novel properties is an outstanding scientific challenge for a broad class of materials that include not only RFEs, but also spin glasses 20 , superelastic strain glasses (shape-memory alloys) 21 , colossal magnetoresistance manganites, and some superconductors 22 .The best-known member of the RFE family is the disordered perovskite crystal PbMg 1/3 Nb 2/3 O 3 (PMN), for which 27 years ago a plausible interpretation of its diffuse-phase-transition (DPT) was proposed by Westphal et al. 3 . Fluctuations of random-internal-electric-field (RIEF) emerging from the quenched charge disorder of the RFE are stabilizing the typical disordered polar nanodomain state. This disordering mechanism convinced sceptical experts at the latest thanks to a favorable review of Cowley et al. 16 . The subseq...