Conditional Value-at-Risk (CVaR) is a widely used risk metric in applications such as finance. We derive concentration bounds for CVaR esti_mates, considering separately the cases of sub_Gaussian, light-tailed and heavy-tailed distribu_tions. For the sub-Gaussian and light-tailed cases, we use a classical CVaR estimator based on the empirical distribution constructed from the samples. For heavy-tailed random variables, we assume a mild 'bounded moment' condi_tion, and derive a concentration bound for a truncation-based estimator. Our concentration bounds exhibit exponential decay in the sample size, and are tighter than those available in the literature for the above distribution classes. To demonstrate the applicability of our concentra_tion results, we consider the CVaR optimization problem in a multi-armed bandit setting. Specifi_cally, we address the best CVaR-arm identifica_tion problem under a fixed budget. Using our CVaR concentration results, we derive an upper_bound on the probability of incorrect arm identi_fication. © International Conference on Machine Learning, ICML 2020. All rights reserved.