This paper presents a conjecture that the duality between color and kinematics, observed at tree level in gauge theories, persists at all quantum loop orders. This duality allows gravity amplitudes to be obtained by taking double copies of gauge-theory diagram numerators. The authors test this conjecture by showing that the three-loop four-point amplitude of N=4 super-Yang-Mills theory can be arranged to satisfy the duality, and by taking double copies of the diagram numerators, they obtain the corresponding amplitude of N=8 supergravity. They also remark on a non-supersymmetric two-loop test based on pure Yang-Mills theory resulting in gravity coupled to an antisymmetric tensor and dilaton. The duality between color and kinematics is conjectured to hold to all multiplicity at tree level in a large variety of theories, including supersymmetric extensions of Yang-Mills theory. This duality surprisingly implies new nontrivial relations between the color-ordered partial amplitudes of gauge theory. A proof of these relations has been made using monodromy for integrations in string theory. The key tool for the studies of loop amplitudes is the unitarity method. An important refinement which simplifies multiloop studies is the method of maximal cuts, which relies on generalized unitarity. The authors use these tools to present an all-loop extension of recently discovered tree-level relations. This allows them to immediately write down multiloop gravity amplitudes directly from gauge-theory multiloop amplitudes once they have been organized to respect the duality between kinematics and color. The authors test their conjecture in a nontrivial case by considering the three-loop four-point amplitude of N=8 supergravity. They show that the four-point three-loop amplitude of N=4 sYM can be organized so its numerator factors satisfy the duality with all internal momenta off shell, and then check if the expression constructed via squaring those numerator factors is the four-point three-loop amplitude of N=8 supergravity. They find that this is indeed the case using a complete set of cuts of the known result. The authors also construct another version of the three-loop four-point N=8 sugra expression using the n_i given in table I of the present Letter and the correct, but duality violating, n_i from table I of ref. [11]. They find that this is also a valid representation of the N=8 sugra three-loop four-point amplitude, providing a strong consistency check on table I and their conjecture. The authors conclude that the gauge-theory duality between color and kinematic numerators imposed in ref. [8] carries over naturally to loop level. This allows the expression of numerators of gravity diagrams using two copies of gauge-theory ones. The known connection between scattering amplitudes of N=4 super-Yang-Mills theory at weak and strong coupling suggests that the duality between colorThis paper presents a conjecture that the duality between color and kinematics, observed at tree level in gauge theories, persists at all quantum loop orders. This duality allows gravity amplitudes to be obtained by taking double copies of gauge-theory diagram numerators. The authors test this conjecture by showing that the three-loop four-point amplitude of N=4 super-Yang-Mills theory can be arranged to satisfy the duality, and by taking double copies of the diagram numerators, they obtain the corresponding amplitude of N=8 supergravity. They also remark on a non-supersymmetric two-loop test based on pure Yang-Mills theory resulting in gravity coupled to an antisymmetric tensor and dilaton. The duality between color and kinematics is conjectured to hold to all multiplicity at tree level in a large variety of theories, including supersymmetric extensions of Yang-Mills theory. This duality surprisingly implies new nontrivial relations between the color-ordered partial amplitudes of gauge theory. A proof of these relations has been made using monodromy for integrations in string theory. The key tool for the studies of loop amplitudes is the unitarity method. An important refinement which simplifies multiloop studies is the method of maximal cuts, which relies on generalized unitarity. The authors use these tools to present an all-loop extension of recently discovered tree-level relations. This allows them to immediately write down multiloop gravity amplitudes directly from gauge-theory multiloop amplitudes once they have been organized to respect the duality between kinematics and color. The authors test their conjecture in a nontrivial case by considering the three-loop four-point amplitude of N=8 supergravity. They show that the four-point three-loop amplitude of N=4 sYM can be organized so its numerator factors satisfy the duality with all internal momenta off shell, and then check if the expression constructed via squaring those numerator factors is the four-point three-loop amplitude of N=8 supergravity. They find that this is indeed the case using a complete set of cuts of the known result. The authors also construct another version of the three-loop four-point N=8 sugra expression using the n_i given in table I of the present Letter and the correct, but duality violating, n_i from table I of ref. [11]. They find that this is also a valid representation of the N=8 sugra three-loop four-point amplitude, providing a strong consistency check on table I and their conjecture. The authors conclude that the gauge-theory duality between color and kinematic numerators imposed in ref. [8] carries over naturally to loop level. This allows the expression of numerators of gravity diagrams using two copies of gauge-theory ones. The known connection between scattering amplitudes of N=4 super-Yang-Mills theory at weak and strong coupling suggests that the duality between color