In this paper we formulate a new structure of spaces which we call it transversal (upper or lower) intervally spaces. We introduce this concept as a natural extension of transversal probabilistic and Menger's spaces. Transversal intervally spaces are a new concept of spaces in the fixed point theory and further a new way in the nonlinear analysis. In this sense, we introduce notions of the intervally contractions on upper and lower transversal intervally spaces and prove some fixed point statements.