In the system of higher education the main results of mastering the basic educational programs are the competences of students and graduates. However, in the pedagogical environment, despite the widespread use of the term, there is still no deep understanding of the features of such results and ways to achieve them. Therefore, there is often no clear distinction between cognitive outcomes (knowledge, skills, and abilities) and competence outcomes. There are difficulties in taking into account both conditions and principles of competence formation and specifics of control and assessment process organization to reliably identify the levels of students’ competence formation. The article gives a brief overview of the existing concepts of competences, their specificity, conditions of their formation and assessment. It is shown that competences reflect students’ abilities to act productively in a problem situation, using knowledge, abilities, and skills to achieve the target result. The intensive development of digital technologies creates additional opportunities and problems not only for distance learning, but also for creating new conditions for assessing learning achievements, including competences, in the virtual environment of digital space. The importance of developing a methodology for competency-based learning, including the use of digital capabilities, is emphasized. It is noted that, despite the many important and necessary tools of the digital space, there are also negative aspects of their use, which should be taken into account in the construction of educational content.