The nuclear Overhauser effect (NOE) is the transfer of nuclear spin polarization from one population of spin-active nuclei (e.g. ¹H, ¹³C, ¹⁵N etc.) to another via cross-relaxation. A phenomenological definition of the NOE in nuclear magnetic resonance spectroscopy (NMR) is the change in the integrated intensity (positive or negative) of one NMR resonance that occurs when another is saturated by irradiation with an RF field. The change in resonance intensity of a nucleus is a consequence of the nucleus being close in space to those directly affected by the RF perturbation.

The NOE is particularly important in the assignment of NMR resonances, and the elucidation and confirmation of the structures or configurations of organic and biological molecules. The ¹H two-dimensional NOE spectroscopy (NOESY) experiment and its extensions are important tools to identify stereochemistry of proteins and other biomolecules in solution, whereas in solid form crystal x-ray diffraction typically used to identify stereochemistry. The heteronuclear NOE is particularly important in ¹³C NMR spectroscopy to identify carbons bonded to protons, to provide polarization enhancements to such carbons to increase signal-to-noise, and to ascertain the extent the relaxation of these carbons is controlled by the dipole-dipole relaxation mechanism.

The NOE developed from the theoretical work of American physicist Albert Overhauser who in 1953 proposed that nuclear spin polarization could be enhanced by the microwave irradiation of the conduction electrons in certain metals. The electron-nuclear enhancement predicted by Overhauser was experimentally demonstrated in ⁷Li metal by T. R. Carver and C. P. Slichter also in 1953. A general theoretical basis and experimental observation of an Overhauser effect involving only nuclear spins in the HF molecule was published by Ionel Solomon in 1955. Another early experimental observation of the NOE was used by Kaiser in 1963 to show how the NOE may be used to determine the relative signs of scalar coupling constants, and to assign spectral lines in NMR spectra to transitions between energy levels. In this study, the resonance of one population of protons (¹H) in an organic molecule was enhanced when a second distinct population of protons in the same organic molecule was saturated by RF irradiation. The application of the NOE was used by Anet and Bourn in 1965 to confirm the assignments of the NMR resonances for β,β-dimethylacrylic acid and dimethyl formamide, thereby showing that conformation and configuration information about organic molecules in solution can be obtained. Bell and Saunders reported direct correlation between NOE enhancements and internuclear distances in 1970 while quantitative measurements of internuclear distances in molecules with three or more spins was reported by Schirmer et al.

Richard R. Ernst was awarded the 1991 Nobel Prize in Chemistry for developing Fourier transform and two-dimensional NMR spectroscopy, which was soon adapted to the measurement of the NOE, particularly in large biological molecules. In 2002, Kurt Wuthrich won the Nobel Prize in Chemistry for the development of nuclear magnetic resonance spectroscopy for determining the three-dimensional structure of biological macromolecules in solution, demonstrating how the 2D NOE method (NOESY) can be used to constrain the three-dimensional structures of large biological macromolecules. Professor Anil Kumar was the first to apply the two-dimensional Nuclear Overhauser Effect (2D-NOE now known as NOESY) experiment to a biomolecule, which opened the field for the determination of three-dimensional structures of biomolecules in solution by NMR spectroscopy.

The NOE and nuclear spin-lattice relaxation are closely related phenomena. For a single spin-1⁄2 nucleus in a magnetic field there are two energy levels that are often labeled α and β, which correspond to the two possible spin quantum states, +1⁄2 and -1⁄2, respectively. At thermal equilibrium, the population of the two energy levels is determined by the Boltzmann distribution with spin populations given by P_α and P_β. If the spin populations are perturbed by an appropriate RF field at the transition energy frequency, the spin populations return to thermal equilibrium by a process called spin-lattice relaxation. The rate of transitions from α to β is proportional to the population of state α, P_α, and is a first order process with rate constant W. The condition where the spin populations are equalized by continuous RF irradiation (P_α = P_β) is called saturation and the resonance disappears since transition probabilities depend on the population difference between the energy levels.

In the simplest case where the NOE is relevant, the resonances of two spin-1⁄2 nuclei, I and S, are chemically shifted but not J-coupled. The energy diagram for such a system has four energy levels that depend on the spin-states of I and S corresponding to αα, αβ, βα, and ββ, respectively. The W's are the probabilities per unit time that a transition will occur between the four energy levels, or in other terms the rate at which the corresponding spin flips occur. There are two single quantum transitions, W₁^I, corresponding to αα ➞ βα and αβ ➞ ββ; W₁^S, corresponding to αα ➞ αβ and βα ➞ ββ; a zero quantum transition, W₀, corresponding to βα ➞ αβ, and a double quantum transition corresponding to αα ➞ ββ.

While rf irradiation can only induce single-quantum transitions (due to so-called quantum mechanical selection rules) giving rise to observable spectral lines, dipolar relaxation may take place through any of the pathways. The dipolar mechanism is the only common relaxation mechanism that can cause transitions in which more than one spin flips. Specifically, the dipolar relaxation mechanism gives rise to transitions between the αα and ββ states (W₂) and between the αβ and the βα states (W₀).

Expressed in terms of their bulk NMR magnetizations, the experimentally observed steady-state NOE for nucleus I when the resonance of nucleus S is saturated (${\displaystyle M_{S}=0}$) is defined by the expression: