The material is divided into six chapters, each of which consists of several sections. Chapter 1 contains definitions and preliminary information on the basic objects in use. The majority of sections of Chapter 2 can also be regarded preliminary, they include general information on continuous Banach bundles. Chapter 3 is central both in position and content: it concentrates the material related to ample Banach bundles and representation of lattice-normed spaces as spaces of sections. In Chapter 4, the theory of measurable Banach bundles is developed, which is made by transferring Daniell's scheme to the case of sections. In the same chapter, the notion of lifting in the space of measurable sections is introduced and studied, and the results are presented of applying the theory of ample Banach bundles to the study of measurable bundles. Chapter 5 contains applications of the results of the preceding chapters to various spaces of continuous and measurable vector-valued functions. Finally, Chapter 6 is devoted to the study of disjointness preserving operators and constructing analytic representations of such operators.