In the past decade, domain engineered relaxor-PT ferroelectric single crystals, including (1-x)Pb(Mg1/3Nb2/3)O3-xPbTiO3 (PMN-PT), (1-x)Pb(Zn1/3Nb2/3)O3-xPbTiO3 (PZN-PT) and (1-x-y)Pb(In1/2Nb1/2)O3-yPb(Mg1/3Nb2/3)O3-xPbTiO3 (PIN-PMN-PT), with compositions near the morphotropic phase boundary (MPB) have triggered a revolution in electromechanical devices owing to their giant piezoelectric properties and ultra-high electromechanical coupling factors. Compared to traditional PbZr1-xTixO3 (PZT) ceramics, the piezoelectric coefficient d33 is increased by a factor of 5 and the electromechanical coupling factor k33 is increased from < 70% to > 90%. Many emerging rich physical phenomena, such as charged domain walls, multi-phase coexistence, domain pattern symmetries, etc., have posed challenging fundamental questions for scientists. The superior electromechanical properties of these domain engineered single crystals have prompted the design of a new generation electromechanical devices, including sensors, transducers, actuators and other electromechanical devices, with greatly improved performance. It took less than 7 years from the discovery of larger size PMN-PT single crystals to the commercial production of the high-end ultrasonic imaging probe “PureWave”. The speed of development is unprecedented, and the research collaboration between academia and industrial engineers on this topic is truly intriguing. It is also exciting to see that these relaxor-PT single crystals are being used to replace traditional PZT piezoceramics in many new fields outside of medical imaging. The new ternary PIN-PMN-PT single crystals, particularly the ones with Mn-doping, have laid a solid foundation for innovations in high power acoustic projectors and ultrasonic motors, hinting another revolution in underwater SONARs and miniature actuation devices. This article intends to provide a comprehensive review on the development of relaxor-PT single crystals, spanning material discovery, crystal growth techniques, domain engineering concept, and full-matrix property characterization all the way to device innovations. It outlines a truly encouraging story in materials science in the modern era. All key references are provided and 30 complete sets of material parameters for different types of relaxor-PT single crystals are listed in the Appendix. It is the intension of this review article to serve as a resource for those who are interested in basic research and practical applications of these relaxor-PT single crystals. In addition, possible mechanisms of giant piezoelectric properties in these domain-engineered relaxor-PT systems will be discussed based on contributions from polarization rotation and charged domain walls.