The whole Thesis is united by the idea of topology and its applications. Nevertheless, we tried to write the chapters in an independent and self-contained way. Chapter 1 gives a general introduction to the topics discussed in this Thesis. Chapters 2 and 4 review the main general concepts that are important for understanding the further results. The results of the research are described in Chapters 3, 5, and 6. Chapters 3 and 5 have the subject of applications of topology in condensed-matter physics. Starting from Chapter 5, we discuss topological solitary waves in classical field theory in various inhomogeneous settings, whereas in Chapter 6 we focus on problems that might have influence in biology or medicine in the future. At the end of the Thesis, the reader will find the bibliography, the summaries in English and Dutch, acknowledgments, and the Curriculum Vitae of the author.