"The project concerns investigations of stochastic recursions, related limit theorems and their applications in branching processes. Recently we have proved many properties of matrix recursions, when the Lyapunov exponent is negative, including precise description of the tail of the stationary measure and resulting limit theorems. We are going to develop further the methods we have used and apply them to study new problems. The main research objectives are:- Studying of stochastic recursions when the consecutive increments are dependent and form a stationary Markov chain. Up to now only the affine recursion has been studied and under restrictive hypotheses, existence of the stationary measure and its tail have been described. We are going to prove related limit theorems and then to investigate matrix recursions and general stochastic recursions.- Description of the invariant measure in the critical case, when the Lyapunov exponent is null. We have studied the case of one dimensional recursions and then we proved regular behavior at infinity of the invariant measure. Now we will concentrate on matrix recursions.- Matrix valued branching processes and Mandelbrot equation. We would like, relying on our experience on affine recursions, to study multidimensional branching processes, where scalars are replaced by positive matrices. We will investigate existence of solutions of the Mandelbrot equation and their asymptotic properties. The problem is important in the context multitype branching processes and random walks on trees in random environments."