When an intense laser beam interacts with matter, the Kerr nonlinearity results in self-focusing. Above the critical intensity, self-focusing dominates pulse spreading through diffraction leading to continuous pulse narrowing and thus an increase of the laser peak intensity. Collapse is prevented through the fact that peak intensities ultimately reach a level where ionization occurs. The profile of ionized electrons represents a negative lens which balances Kerr nonlinear self-focusing and causes the formation of stable filaments. From filaments radiation is emitted in a cone around the filament which has been termed conical emission. Filament formation happens at non-perturbative intensities where the formalism of perturbative nonlinear optics loses its validity. This opens the question of how the Kerr nonlinearity behaves in the non-perturbative limit and how large the Kerr nonlinear coefficient is. The expression for the Kerr nonlinearity is derived by perturbation theory; the validity of this expression in the non-perturbative limit is questionable. Further, experimentally the Kerr nonlinear index is extracted from measurements of the self-focusing distance as a function of beam intensity which is called the Z-scan method. This method fails at non-perturbative intensities due to the presence of the negative lens coming from the ionized electrons. The effects of the positive focusing and negative self-defocusing lens cannot be separated by the Z-scan method. As a result, not much is known about the Kerr nonlinearity in the regime of non-perturbative nonlinear optics. The purpose of this thesis is twofold. First, recently it has been discovered that conical emission can be utilized as a broadband and very efficient amplification mechanism in the far infrared. The process has been dubbed Kerr instability amplification. The difference between conical emission and Kerr instability amplification is that they take place in two different regimes of the nonlinear interaction. Whereas conical emission grows out of noise and therewith only takes place once the pump pulse has been substantially restructured due to filamentation, Kerr instability amplification is seeded with a second pulse and therewith occurs long before filamentation happens. The theory developed for Kerr instability amplification has been developed based on a stability analysis of the scalar wave equation. This analysis has shown that with pump lasers in the 1-2 μm range amplification of infrared radiation up to the 10’s of μm can be achieved. For amplification over such a wide range it is not adhoc clear to which exent vectorial wave effects can be neglected. The first part of the thesis closes this gap by developing the vectorial theory of vector instability amplification. The second part uses the results derived for Kerr instability amplification to answer the question of how to measure the Kerr nonlinear index in the nonperturbative laser intensity limit. The idea rests on the fact that Kerr instability amplification is maximum for a specific angle between pump and seed beam which varies as a function of laser pump intensity. A relation is derived that connects this angle with the Kerr nonlinear refractive index. As a result, from the maximum angle measured as a function of pump intensity, both magnitude and functional form of the Kerr nonlinear index as a function of laser intensity can be determined.