Fundamentals of the mathematical theory of growing solids are under consideration. The theory of fiber bundles of differentiable manifolds is taken as its foundation. Classification of the possible ways of solids accretion depending on the dimension of fiber (or dimension of fiber bundle base) is given. Accretion of 3D solids by 2D surfaces is under detailed investigation. The basic system of equations as well as the boundary and complimentary conditions of the considered process of solid accretion are presented. They fundamentally differ from the equations and conditions of the classical case. The specific features of the proposed mathematical theory, in particular, its geometric aspects are discussed.
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