Abstract: The mathematical field of Selection Principles deals with qualitative analysis of diagonalization procedures, where the diagonalized object carries a mathematical (topological or other) structure.

This field aims at applications in a variety of mathematical disciplines, including: Ramsey theory, game theory, function spaces and convergence, topological groups, dimension theory, covering properties, combinatorial set theory, hyperspaces, etc.

The uniform treatment given by Selection Principles to this variety of fundamental mathematical properties is not only aesthetically pleasing. It also brought new perspectives, focused attention on essential parts of classical problems, and in several cases lead to their solution.

We will illustrate this by several fundamental examples to which we have contributed in the last few years, and discuss some of our future plans regarding this fascinating topic.