In this thesis the following problems are studied: 1-Relativistic self-focusing and channelling of intense laser pulses have been studied in underdense plasma using 2D PIC simulations, for different laser powers and plasma densities. Analytical solutions for the stationary evacuated channels have been recovered in PIC simulations. It is shown that otherwise stable channels can accelerate electrons due to surface waves on the walls of the channels. Relativistic filaments with finite electron density are unstable to transverse modulations which lead in the nonlinear stage to the break-up of laser pulses into independent filaments. 2-Although 3D simulations are limited, they are more realistic. Azimuthal stability of the laser pulses in interaction with underdense plasma can only be studied in 3D geometry. Relativistic self-focusing and channelling of intense laser pulses have been studied in underdense plasma using 3D PIC simulations, for different laser powers and plasma densities. Analytical solutions for the stationary evacuated channels and ring structure have been recovered in PIC simulations. The stability of ring structure due to azimuthal perturbations has been studied both in theory and in simulations. The gain length of such instability is smaller at higher densities $(>0.1n_{cr})$. It is shown that the azimuthal perturbation can break up the azimuthal symmetry of the laser pulse. 3-Working with circularly polarized laser pulses, gave us a motivation to study Inverse Faraday Effect in interaction of circularly polarized laser pulses with plasma. Axial magnetic field generation by intense circularly polarized laser beams in underdense plasmas has been studied with 3D particle-in-cell (PIC) simulations and by means of theoretical analysis. The source of azimuthal nonlinear currents and of the axial magnetic field depends on the transverse inhomogeneities of the electron density and laser intensity. The fields reach maximum strength of several tens of MG for laser pulses undergoing relativistic self-focusing and channelling in moderately relativistic regime. 4-Electron wakefield acceleration was studied in support of the experiment which was carried on using 7 TW laser beam at Canadian Advanced Laser Light source facility. 2D simulations were performed to study this problem. The energy the electrons gained in the process was peaked at 20-30 Mev close to the experimental results.

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