Constructing skew and heavy-tailed distributions by transforming a standard normal variable goes back to Tukey (1977) and was extended and formalized by Hoaglin (1983) and Martinez & Iglewicz (1984). Applications of Tukey's GH distribution family - which are composed by a skewness transformation G and a kurtosis transformation H - can be found, for instance, in financial, environmental or medical statistics. Recently, alternative transformations emerged in the literature. Rayner & MacGillivray (2002b) discuss the GK distributions, where Tukey's H-transformation is replaced by another kurtosis transformation K. Similarly, Fischer & Klein (2004) advocate the J-transformation which also produces heavy tails but - in contrast to Tukey's H-transformation - still guarantees the existence of all moments. Within this work we present a very general kurtosis transformation which nests H-, K- and J-transformation and, hence, permits to discriminate between them. Applications to financial and teletraffic data are given.