Galton–Watson processes and their role as building blocks for branching processes

This article is an essay, both expository and argumentative, on the Galton–Watson process as a tool in the domain of branching processes. It is at the same time the author's way of honoring two distinguished scientists in this domain, both from the Russian Academy of Sciences, and congratulating them on their special birthdays. The thread of the article is the role which the Galton–Watson process had played in the author's own research. We start with a discussion on a controlled Galton–Watson process. Then we pass to random absorbing processes, and also recall and discuss a problem in medicine. Further questions bring us, via the Borel–Cantelli lemma, to $\varphi$-branching processes and extensions. To gain more generality we then look at bisexual Galton–Watson processes. Finally, we briefly discuss relatively complicated resource-dependent branching processes to show that, again, using Galton–Watson reproduction schemes (whenever reasonable) can be a convincing approach to new processes, which are then sufficiently tractable to obtain results of interest.