This habilitation thesis is organized in two independent parts: a first part about my main results on Steinhaus triangles and graphs and a second part about other results obtained on numerical semigroups, Tower of Hanoi problems, Ramsey theory and Kneser transversals. The topic of the first part, Steinhaus triangles and graphs, is my oldest research theme, starting with my doctoral thesis, that I defended in 2008. Since then, I have continued to work on and explore problems on these structures and on generalizations. This part is divided into five chapters: two chapters about binary Steinhaus triangles, the first chapter on balanced triangles and the second one on Steinhaus triangles having symmetric properties, one chapter on Steinhaus graphs and finally two chapters on generalizations of binary Steinhaus triangles, one for a generalization with modular numbers and one other for generalizations in higher dimensions with balanced simplices. The second part of this habilitation thesis is divided into four chapters. Each chapter is about a research topic on which I began to work after my doctoral thesis: numerical semigroups, Tower of Hanoi problems, Ramsey theory and Kneser transversals.