Weakly linear systems of fuzzy relation inequalities and equations have recently emerged from research in the theory of fuzzy automata. From the general aspect of the theory of fuzzy relation equations and inequalities homogeneous and heterogeneous weakly linear systems have been discussed in two recent papers. Here we give a brief overview of the main results from these two papers, as well as from a series of papers on applications of weakly linear systems in the state reduction of fuzzy automata, the study of simulation, bisimulation and equivalence of fuzzy automata, and in the social network analysis. Especially, we present algorithms for computing the greatest solutions to weakly linear systems.