Cloud model is a cognitive model which can realize the bidirectional cognitive transformation between qualitative concept and quantitative data based on probability statistics and fuzzy set theory. It uses the forward cloud transformation (FCT) and the backward cloud transformation (BCT) to implement the cognitive transformations between the intension and extension of a concept. As one of the most important cloud models, the normal cloud models, especially the 2nd-order normal cloud model based on normal distribution and Gaussian membership function has been extensively researched and successfully applied to many fields. In this paper, a 2nd-order generic normal cloud model, which establishes a relationship between normal cloud and normal distribution, is proposed, and the 2nd-order generic forward normal cloud transformation algorithm (2nd-GFCT) is presented. Whereafter, an ideal backward cloud transformation algorithm of the 2nd-order generic normal cloud model (2nd-GIBCT) is designed based on the mutually inverse features of FT and BCT, in which the distribution of all the cloud drops generated in 2nd-GFCT is used. Meanwhile, a 2nd-order generic backward cloud transformation algorithm (2nd-GBCT), which does not use the distribution of cloud drops, is also proposed to solve real life problems since it is impossible to know the distribution of all the cloud drops in advance in real life applications. The relationships between the generic backward cloud transformation algorithms are further studied, which help reach the finding that the two backward cloud transformation algorithms presented by Wang and Xu [26,34] are two special cases of the 2nd-GBCT. In addition, the 2nd-order generic normal cloud model is further generalized to pth-order generic normal cloud model, and the pth-order generic forward normal cloud transformation algorithm (pth-GFCT) and the backward cloud transformation algorithm (pth-GBCT) are presented. Finally, the performances of the 2nd-GIBCT and the 2nd-GBCT are illustrated by simulation experiment. The effectiveness of the 2nd-GBCT is shown by the results of image segmentation. (C) 2014 Elsevier Inc. All rights reserved.