A vector that is a conserved quantity in the elliptical motion of a single body orbiting another body where there is an inverse-square law of attraction between the two bodies. In the case of celestial mechanics, the physical significance of the Runge–Lenz vector being conserved is that it indicates that the ellipse of the orbit is fixed and does not precess.

There is an analogue of the Runge–Lenz vector in quantum mechanics, which Wolfgang Pauli used to solve the problem of the hydrogen atom. It was subsequently realized that the conservation of the Runge–Lenz vector in the hydrogen atom is associated with a four-dimensional symmetry, giving rise to the Fock degeneracy. Although it was known much earlier, the Runge–Lenz vector is named after Carl Runge (1856–1927), who publicized it in a book on vectors in 1919, and Wilhelm Lenz (1888–1957), who used it in an extension of the Bohr theory in 1924.