[en] Nuclear Magnetic Resonance (NMR) and Nuclear Quadrupole Resonance (NQR) are versatile and very successful spectroscopic methods to investigate, on the atomic scale, the dynamics of cuprate high-temperature superconductors like ,YBa₂Cu₃O₇, YBa₂Cu₄O₈ and Y₂Ba₄Cu₇O₁₅. The studies were performed in both the normal and the superconducting state thereby using various NMR isotopes (e.g. ¹⁷ O, ⁶³ Cu, ¹³⁵ Ba). Here, we will focus on studies of the spin-lattice relaxation time which yields, among others, information about the type of coupling between the nucleus and its surrounding (i.e. magnetic or quadrupole coupling) and about the mechanism causing the fluctuations (e.g. antiferromagnetic fluctuations, phonons, and perhaps polarons). The dynamical spin susceptibility, χ(q vector, ω) depending on wavevector and frequency, is the starting point of the discussion. Various models have been proposed to determine χ(q vector, ω), for instance the famous phenomenological Millis-Monien-Pines model; the relaxation rate can be calculated once χ(q vector, ω) is known. We will also mention in some detail two other theories. The first is based on the t-J model, presented in terms of Hubbard operators (where t and J denote the hopping energy and the repulsion of holes, respectively), which is applied to plane nuclei like copper and oxygen. The other is an application of the Luttinger-liquid model for describing one-dimensional quantum systems; this model is well suited to deal with the dynamics of copper nuclei in the chains of YBa₂Cu₃O₇ and YBa₂Cu₄O₈. Further topics will be the possible evidence for a charge density wave in YBa₂Cu₄O₈, the effect of phonons on out-of-plane nuclei like Ba, and the interplane coupling and spin transfer in Y₂Ba₄Cu₇O₁₅. Finally, we will deal with some aspects of the superconducting phase: the spin-singlet pairing, the d-wave pairing of the charge carriers and an unusual relaxation phenomenon at very low temperatures. (author)