The book contains a systematic exposition of the properties of stable distributions, limit theorems on convergence to them, limit theorems for the probabilities of large deviations of sums of independent equally distributed random variables with distributions from the attraction domains of asymmetric stable distributions and the normal law, as well as limit convergence theorems with probability one for the mentioned sums and their increments. The book covers both a number of well-known issues and a number of topics that have not previously been addressed in educational literature.