Linear degenerations of the flag variety arise as natural generalizations of the complete flag variety, and their geometric properties frequently reveal intriguing combinatorial patterns. In the first part of this talk, I will introduce these objects and discuss how the celebrated Schubert cells decomposition of the complete flag variety can be generalized in this context. Moreover, I will introduce basic notions about quiver representation theory and discuss how linear degenerations can be described as varieties of representations of equioriented quivers of type A. In the second part, I will provide an overview of their geometric properties and outline the main open problems in this interesting research area. Finally, I will present some very recent general results about the smoothness and irreducibility of these varieties.