The proposed project aims at analyzing fundamental problems from combinatorics using the most current methods available and at providing new structural and algorithmic insights to such problems. The problems considered will be treated on a general level of classes of combinatorial objects of the same kind and the developed general methods will also be applied to specific open problems. Classes of dense and sparse objects will be treated using different techniques. Dense combinatorial objects appear in extremal combinatorics and tools developed to handle them found their applications in differentareas of mathematics and computer science. The project will focus on extending known methods to new classes of combinatorial objects, in particular those from algebra, and applying the most current techniques including Razborov flag algebras to problems from extremal combinatorics. Applications of the obtained results in property testing will also be considered. On the other hand, algorithmic applications often include manipulating with sparse objects. Examples of sparse objects are graphs embeddable in a fixed surface and more general minor-closed classes of graphs. The project objectives include providing new structural results and algorithmic metatheorems for classes of sparse objects using both classical tools based on the theory of graph minors as well as new tools based on the framework of classes of nowhere-dense structures.