Abstract: We consider sums of n i.i.d. random variables with tails close to exp{−x^β} for some β > 1. Asymptotics developed by Rootzén (1987) and Balkema, Klüppelberg, and Resnick (1993) are discussed from the point of view of tails rather than of densities, using a somewhat different angle, and supplemented with bounds, results on a random number N of terms, and simulation algorithms.