The article deals with the problem of designing new educational results using the tools of a multi-environment lesson, among which the author highlights and defines cross-cutting competencies. The author's classification of cross-cutting competencies is also presented. With regard to the school stage of human education, seven generalized groups of competencies are proposed: social; educational and cognitive; psychological - pedagogical; general civil; technological; general cultural; managerial. The problems of a multi-environment lesson, classes, activities in the article are considered within a broader context associated with the cross-multidimensional organization of the educational process, didactics and organizational and pedagogical approaches to building a multi-environment lesson. The cross-multidimensional organization of the educational process is considered as a result of repetitive and situational, controlled and stochastic interactions and intersections of the educational environment with a variety of spaces and environments, including digital, cultural-historical, ethno-cultural, innovative, technological, valeological, subject-spatial environments; biblio - media, axiological, semantic, semiotic, noospheric, event and other spaces. Examples of intersection and interaction within the lesson, classes on extracurricular activities of individual spaces and environments, as well as the pedagogical results and effects achieved as a result of this are given. A significant place in the article is given to the description of the construction and application of the multi-environment lesson constructor, which allows you to simultaneously use the potentials and capabilities of several spaces and environments, solve educational problems in accordance with the Federal State Educational Standards and the needs of students, ensure the formation of their meta-subject results, cross-cutting competencies, skills of the future. A separate issue, which is covered in the article, is the problem-activity direction of a multi-environment lesson, the role of the teacher in its design, including determining educational results.