  It is a common practice to describe branching random walks in terms of birth,
death and walk of particles, which makes it easier to use them in different
applications. The main results obtained for the models of symmetric
continuous-time heavy-tailed branching random walks on a multidimensional
lattice. We will be mainly interested in studying the problems related to the
limiting behavior of branching random walks such as the existence of phase
transitions under change of various parameters, the properties of the limiting
distribution and the survival probability of a particle population. Emphasis is
made on the survival analysis. The answers to these and other questions
essentially depend on numerous factors which affect the properties of a
branching random walk. Therefore, we will try to describe how the properties of
a branching random walk depend on such characteristics of the underlying random
walk as finiteness or infiniteness of the variance of jumps. The presented
results are based on the Green's function representations of the transition
probabilities of a branching random walk.
