Alien chromatin (introgressions) can be incorporated into the wheat genome either through recombination or through translocation. Lines of introgressive origin may not contain identified alien chromatin, although they appear to be changed relative to the original wheat sample by some characteristics that are registered at different levels of phenotype assessment. It has been shown numerous facts of the variability of the recipient genome of hybrid origin. They can be caused by irregular behavior of chromosomes in mitoses of somatic cells with hybrid genome, and DNA changes. Wide phenotypic variability was observed among the progeny of hybrid genomes up to appearance of phenotypic novelties. The study of DNA profiles using several methods revealed the presence of numerous genetic and epigenetic changes, including deletions, changes in gene expression, changes in the methylation pattern, and activation of retrotransposons in the recipient genome. When these changes were not directly related to alien fragments, they are considered as possible mechanisms of extension of genetic variability intra genome of hybrid origin. Transcriptomic analysis of lines of hybrid origin and parental plants using the RNA-seq analyses showed that the change of expression affected a different number of genes. The change in the expression of alien genes occurred in the direction of down-regulation, upregulation was typical for wheat genes. Phenotypic variability during introgressive hybridization can be accompanied by attenuation of the expression of an alien gene when it enters the recipient genome. That may be a consequence of the interaction between the alien gene and the genetic environment of the host genome. Today there is a lot of evidence that these mechanisms are epigenetic, which include DNA methylation, histone modification, the involvement of small RNAs in the regulation of gene expression, and the involvement of transposons. Moreover, none of these mechanisms act independently, but are a component of events, the results of which can be a change in gene expression and the appearance of a new phenotype that is not related to the expression of an alien gene.