The article analyzes the problems of mathematical university preparation of specialists in the context of their professional self-realization in the modern labour market. The structure of mathematical competence of students is considered, its most important components are substantively described. As a result of pedagogical experiment, the level of students' mathematical competence formation according to the specified structure was established. It is proved that students of both mathematical and non-mathematical majors have poor mathematical knowledge and skills for solving problems that require their complex application, in particular, modeling problems in the context of future professional activity. The most important factors that significantly inhibit the process of learning mathematics at the university are highlighted. The basic approaches to formation of mathematical competence of students are substantiated: research-oriented teaching; interdisciplinary approach and practical orientation; use of digital learning environments and related tools. The methods of implementation of these approaches are disclosed in the authors' elective discipline for the students of mathematical specialty "Methods of formation of students’ mathematical competence for the future successful career". The purpose of this discipline is to improve the methodological preparation of masters for teaching mathematical disciplines in educational establishments.