Abstract: A probability measure $\mu$ on the general linear group $\GL_d(\R)$ induces a non-commutative random walk $R_n$ on $\GL_d(\R)$ and a Markov chain on the projective space $P(\R^d)$. The associated stationary measures on $P(\R^d)$ encodes various information on $R_n$ and their understanding is crucial for the study of asymptotic properties of the random walk. While the pioneering works of Furstenberg, Kifer, Guivarc'h, Hennion etc. have given a satisfactory description of stationary measures (particularly when the measure $\mu$ is irreducible), many natural questions remain to be studied in the reducible case. In this ongoing series of works, we give a description of stationary measures which refines that of Furstenberg--Kifer and Hennion from '80s and also generalizes recent work by Aoun--Guivarc'h and Benoist--Bruère. After reviewing the well-known aspects of the theory, we will discuss our results, techniques and further consequences. Joint work with Richard Aoun.