My thesis deals with two important topics of Cosmology:(i) Origin of cosmological magnetic fields:Magnetic fields seem ubiquitous in the Universe, present at all scales and all times, probably even in the entire intergalactic medium. Their origin is still unclear, especially on the largest scales. The current paradigm is that they were first generated with extremely weak strengths, and later amplified during structure formation. Because of turbulence, the fields we observe in galaxies and galaxy clusters lost their initial characteristics. However, in less dense regions such as cosmological filaments, sheets or voids, magnetic fields have evolved more mildly. Therefore, intergalactic magnetic fields may still possess a memory of the processes that generated them and hold the key to their origin. I developed analytically a detailed physical model of a natural astrophysical mechanism that generates intergalactic magnetic fields during the first billion year, namely at the time when first stars and galaxies were born. Then, in collaboration with H. Tashiro and N. Sugiyama (Japan), I computed analytically the mean energy density injected in the entire Universe through this mechanism. Independently, in collaboration with D. Aubert (France), I derived the topological and statistical properties of the magnetic field thus generated, using cosmological numerical simulations. This way I demonstrated that this simple, natural photoionization-based magnetogenesis must have created magnetic seed fields with properties a priori perfectly compatible with present day observations.(ii) Gravitational fragmentation of the cosmic web:Cosmological numerical simulations suggest that the Universe has a web-like structure, the nodes of which are galaxy clusters. These nodes are supplied with matter flowing along the filaments interconnecting them. Part of this accretion occurs intermittently, which indicates that clumps of matter form not only inside clusters themselves, but also either in cosmic voids, walls and/or filaments. I studied gravitational instability in stratified media in the frame of spectral theory, in planar and cylindrical geometries, relevant for cosmic walls and filaments, for isothermal, polytropic, and with and without an external gravitational background (e.g. Dark Matter). I have recasted the problem as an eigenvalue problem in the force operator formalism, and derived the wave equation governing the growth of perturbations. I also studied it in matrix form, which gives complementary information.