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The phenomenon of laser-induced periodic surface structures (LIPSS), or ripples, generated by near-infrared radiation with the central wavelength around 800 nanometer (λlaser ≈ 800 nm) and pulse duration about of 100 femtosecond (τpulse ≈ 100 fs) on solid targets is considered in this dissertation. The main aim of the work is a better understanding of the fundamental processes of laser-matter interaction resulting in pattern formation by femtosecond (fs) laser ablation. The problem is of great interest both in fundamental and applied science. The knowledge of the underlying physical mechanisms will provide the opportunity to control surface nanostructuring, which has a big application potential in many modern technologies. Femtosecond LIPSS observed at the bottom of ablation crater reveal a large variety of features including nanostructures with periods below 100 nm. Moreover, the ripple size depends mainly on the irradiation dose/absorbed laser energy and is rather insensitive to the variation of laser wavelength or incidence angle. The orientation of the structures is dictated by laser polarization. All these experimental observations and an astounding similarity of the structures to other patterns originating from instabilities led to the idea to attribute the femtosecond laser nanostructuring to a self-organized pattern formation from laser-induced surface instability. In this dissertation, surface pattern formation upon femtosecond laser ablation is considered in the framework of an adopted surface erosion model, based on the description of spontaneous pattern formation on surfaces bombarded with high-energy ions. We exploit the similarity to ion-beam sputtering and extend a corresponding model for laser ablation by including laser polarization. It has been found that an asymmetry in the deposition and dissipation of the incident laser energy, related to the laser polarization, results in a corresponding dependence of coefficients in a nonlinear equation of the Kuramoto-Sivashinsky type. The surface morphologies calculated in the framework of this model for different configurations of the incident laser electric field show an excellent qualitative agreement with structures observed in ultra-short pulse ablation experiments. In this work, properties of the periodic surface structures induced upon femtosecond laser ablation are studied in detail, focusing on a systematic investigation of the main control parameters regulating the pattern formation process. The results support the non-linear self-organization mechanism of pattern formation from laser-induced surface instability.