It is well known that the cylindrical Brownian motion and the Gaussian space-time white noise correspond to each other. In this talk we consider the analogue relation between cylindrical Lévy processes and Lévy space-time white noises. In contrast to the Gaussian case, it turns out that this correspondence only holds for specific kinds of cylindrical Lévy processes. We exploit the established relation by embedding cylindrical Lévy processes in certain Besov spaces, which may be seen as a first result explaining the regular (or irregular) behaviour of the jumps of a cylindrical Lévy process.