In this thesis we have studied the limit behavior of Galton–Watson processes. Such processes describe the evolution of the population which starts with one individual. All individuals reproduce independently from each other and have the same offspring distribution. We show that these processes have a certain dichotomy: they either become extinct or population size explodes. Depending on mean of the offspring distribution m, Galton–Watson process is called subcritical if m < 1, critical if m = 1, and supercritical if m > 1. Subcritical and critical processes become extinct a.e., while supercritical processes survive with positive probability. Finer question is how fast subcritical and critical processes become extinct and what is the growth rate of supercritical processes if they survive. We show that the main question concerning subcritical and supercritical processes is when does \(m^n\) give the right decay rate and the right growth rate respectively. In both cases, it turns out that this is equivalent to ”L log L” criterion which states \(\mathbb{E} [L\log^+ L]<\infty\), where L is a random variable denoting the number of children of some individual. If this condition isn’t true, subcritical processes die out faster than \(m^n\), while the growth rate of supercritical processes is less than \(m^n\). Moreover, in that instance supercritical processes grow just slightly slower than \(m^n\). The behavior of critical processes is somewhat different. They die out with the rate of \(1/n\) if \(Var(L)<\infty\), whereas the pace of extinction is even greater if \(Var(L)=\infty\).