The classification of finite objects (for example, associative/Lie/Poisson/Hopf algebras of a given finite dimension) is an old and notoriously difficult problem. Having the extension problem and the factorization problem as sources of inspiration we have introduced and begun an extensive study program focusing on two problems, namely the Extending Structures (ES) problem and the Global Extension (GE) problem, which are the specific objectives of the present research project. The ES problem can be stated as follows: Let O be a mathematical object in a given category C (e.g. the category of Lie/associative/Hopf/Poisson algebras) and consider c to be a fixed cardinal. Describe and classify all objects of the category C which contain and stabilize O as a subobject of codimension c. Beyond its elementary statement, the problem is very difficult but also very tempting, given the numerous subsequent problems it raises. The GE problem can be stated as follows: Let A be a fixed object in a certain category C (for instance, the category of Lie/associative/Hopf/Poisson algebras etc.) and c a given cardinal. Describe and classify, up to an isomorphism that co-stabilizes A, all objects X in the category C for which there exists an epimorphism f: X→A whose kernel has dimension c. Like the ES-problem, the GE-problem is also very general and has a large number of subsequent problems and applications, such as the classification of  co-flag objects of a given finite dimension.