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This dissertation addresses the important question of the extent to which random variables and vectors with truncated power tails retain the characteristic features of random variables and vectors with power tails. We define two truncation regimes - soft truncation and hard truncation, based on the growth rate of the truncating threshold. We study the central limit theorem and the large deviations behavior of the model with truncated power laws in both regimes. The central limit theorem is studied for random vectors taking values in a separable Banach space, while for the large deviations, the random vectors are assumed to be Rd -valued. It turns out that, in the soft truncation regime, truncated power tails behave, in important respects, as if no truncation took place. On the other hand, in the hard truncation regime much of "heavy tailedness" is lost. Based on this observation, we set before ourselves two tasks. The first one is to suggest statistical tests to decide on whether the truncation is soft or hard. The second task is to devise an estimator for the tail exponent from the truncated data, which is consistent regardless of the truncation regime. Finally, we apply our methods to two recent data sets arising from computer networks.