Since the turn of the century, there has been increased interest in the application of heavy-tailed distributions, particularly stable distributions, to problems in physics and finance. Although, the tails of stable distributions provide a better fit to real-world data, they are too fat to describe empirical distributions. To remedy this drawback of stable distributions, so-called tempered variants of stable distributions have been proposed. In this paper, we argue that the tempering should be connected to the model leading to heavy tailed distribution and propose several tempering procedures in connection with corresponding models