Abstract. We present evidence that the antiferromagnetic spin correlations in optimally doped YBa 2 Cu 3 O 7 and underdoped YBa 2 Cu 4 O 8 develop a surprisingly strong anisotropy in the superconducting state. Comparing the ratio of the nuclear spin-lattice relaxation rates of the planar copper and oxygen, measured at the lowest and highest temperatures as well as at Tc, we conclude that the antiferromagnetic in-plane correlations vanish as the temperature goes to zero. This observation is corroborated by the measurement of the copper linewidth in YBa 2 Cu 4 O 8 . In contrast, the out-of-plane correlations do not change appreciably between T = Tc and T = 0. Within a model of fluctuating fields this extreme anisotropy of the antiferromagnetic correlations also explains the observed temperature dependence of the anisotropy of the copper relaxation measured in a low external magnetic field.Nuclear magnetic/quadrupole resonance (NMR/NQR) experiments have shed light on the antiferromagnetic (AFM) spin correlations in the normal state of hightemperature superconductors. It was found that the correlations become progressively stronger as the temperature is reduced towards the superconducting transition temperature T c . The question is to what extent these correlations persist below T c . As the temperature is lowered in the superconducting state, the spin-lattice relaxation1α of the planar copper (k = 63) and planar oxygen (k = 17) nuclei decrease rapidly, with the applied field either parallel to the CuO 2 plane (α = ab) or perpendicular to it (α = c). This decrease goes approximately like T 3 , which is the temperature dependence expected for d-wave orbital pairing [1]. Great emphasis has been put on the different temperature behaviour of the copper and oxygen relaxation rates in the normal state. However, less attention has been payed on the ratios of these rates [2][3][4][5][6][7][8][9] in the superconducting state. A difficulty arises since in the normal state, the spin-lattice relaxation is largely insensitive to the strength of the applied magnetic field [10,11] [5][6][7]12]. In order to draw any conclusions about magnetism in the superconducting state it is therefore very important to look for intrinsic effects which can only be obtained from experiments done in weak magnetic fields so as to minimise the flux line influence. We will consider four NMR/NQR experimental results: the ratio 63 T