<p>We present a survey of some author’s results related to the development of the Lévy analysis in the spaces of irregular test and generalized functions. The indicated spaces are constructed by using Lytvynov’s generalization of the chaotic representation property, which is a direct analog of the decomposition of squareintegrable random variables in the Hermite orthogonal polynomials in the Gaussian analysis. In this approach, numerous definitions and statements are quite similar to their prototypes from the Gaussian analysis, which is very convenient for applications. The survey covers a fairly broad range of issues, namely, the extended stochastic integral, the Hida stochastic derivative, their generalizations, the operators of stochastic differentiations and their analogs (generalizations), elements of the Wick calculus, the relationship between the Wick calculus and integration, etc.</p>