The study considers problem-based assignments of spiritual and cognitive characters contributing to the development of axiological competence of students learning a foreign language. The undertaken analysis is aimed at justifying the objective necessity of including the axiological competence into key competences developed during the educational process, and the possibility of developing such a competence within the problem-based training of a foreign language, where problematic assignments of spiritual and cognitive character can be the means of creating the axiological competence. The analysis has shown that the axiological competence provides for the mechanism for self-identity and preconditions the educational trajectory of an individual and his or her life program as a whole. This competence is manifested in actions, attitudes, and activities of an individual. The author has identified the relation between the content of the competence with the content of general personal culture and distinguished the characteristics and skills of the axiological competence. The article reveals the educational potential of problem-based training of a foreign language manifesting itself in three aspects, such as developing a culture of thinking, a culture of communication and a culture of a value-based attitude to reality. The study also considers a system of problem-based assignments and tasks when working with texts with axiological problems. The author proves that the discussion of topical spiritual and cognitive problems in the process of learning a foreign language helps to develop axiological competence and general culture of a student. The system of problem-based assignments can be used in teaching the German language.