The textbook is the basis of the courses "Mathematical Foundations of Continuum Mechanics" and "Continuum Mechanics Models," as well as elective courses at universities where there is a specialization of students, undergraduates and graduate students in the field of natural and technical sciences. It gives a synthesis of the algebraic and geometric description of the tensor apparatus, its application to the results of differential geometry often used in mechanics and physics, to the construction of closed models of continuum mechanics.