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High entropy alloys are multicomponent alloys, which consist of five or more elements in equiatomic or nearly equiatomic concentrations. These materials are hypothesized to show significantly decreased self-diffusivities. For the first time, diffusion of all constituent elements in equiatomic CoCrFeNi and CoCrFeMnNi single crystals and additionally solute diffusion of Mn in the quaternary alloy is investigated using the radiotracer technique, thereby the tracer diffusion coefficients of 57Co, 51Cr, 59Fe, 54Mn, and 63Ni are determined at a temperature of 1373 K. The components are characterized by significantly different diffusion rates, with Mn being the fastest element and Ni and Co being the slowest ones. Furthermore, solute diffusion of Cu in the CoCrFeNi single crystal is investigated in the temperature range of 973–1173 K using the 64Cu isotope. In the quaternary alloy, Cu is found to be a fast diffuser at the moderate temperatures below 1273 K and its diffusion rate follows the Arrhenius law with an activation enthalpy of about 149 kJ/mol.