The work is devoted to order properties and analytic representation of some classes of linear and nonlinear (multilinear, polynomial, sublinear) operators in vector lattices and Banach section spaces. The presentation is concentrated mainly around three topics: envelope representation method and general inequalities in uniformly complete vector lattices; order structure and analytic representation of multilinear operators and polynomials; dominated operators in Banach lattices of continuous and measurable sections associated with the continuous or measurable bundles of Banach lattices. The aim of this collective work is to give an overview of main results obtained last three years within the framework of the project “Operators in spaces of sections of measurable Banach bundles”. The material in the paper is based upon work supported by Russian Foundation of Basic Researches under Grant № 09-01-00442-a.