Abstract: One of the central problems of statistical learning theory is quantifying the generalization ability of learning algorithms within a probabilistic framework. Algorithmic stability is a powerful tool for deriving generalization bounds, however, it typically builds on a critical assumption that losses are bounded. In this paper, we relax this condition to unbounded loss functions with subweibull diameter. This gives new generalization bounds for algorithmic stability and also includes existing results of subgaussian and subexponential diameters as specific cases. Furthermore, we provide a refined stability analysis by developing generalization bounds which can be $\sqrt{n}$-times faster than the previous results, where $n$ is the sample size. Our main technical contribution is general concentration inequalities for subweibull random variables, which may be of independent interest.