In this thesis, we have investigated the electrostatic electron Bernstein waves in a collisionless magnetized plasma using the Cairns distribution function. In this regard, we have derived a generalized dielectric constant for the Bernstein waves and derived the modified dispersion relation in the presence of Cairns distribution function. We have found that the dispersion curves for the electron Bernstein waves using the Cairns distribution function show a very significant deviation from the Maxwellian results. It has been found that the behavior of the Bernstein waves across the entire band between the adjacent harmonics show a departure from the Maxwellian result for the different values of the non-thermality parameter for the Cairns distribution function. Furthermore, Bernstein waves with Vasyliunas-Cairns (VC) distribution has been investi gated. We have obtained the linear dispersion relation for Bernstein waves for two electron species and explored it both analytically and numerically. It has been shown that Bernstein waves with Vasyliunas-Cairns distribution exhibit a significant departure from the results re ported earlier with Maxwellian and dual kappa distributions especially in the long wavelength regime. It has been found that the parameters represent the non-thermal percentage of cold and hot electrons in Cairns distribution, play a unique and vital role in determining the behavior of Bernstein modes in VC distribution which has no parallel in Maxwellian and double kappa distribution functions. Whistler waves propagating parallel to the ambient magnetic field with electron temperature anisotropy have also been investigated employing the kinetic theory of plasmas. The electron distribution function has been considered to follow the Cairns distribution. Dispersion relation for the whistler waves with Cairns distribution has been derived and the condition for the onset of instability has also been obtained. It has been found that the Cairns distribution significantly modifies the instability condition for the growth of whistler instability. The comparison of the dispersion characteristics and growth rate with Maxwellian distribution has also been made and it has been observed that Cairns distributed electrons yield a higher growth rate by comparison with their Maxwellian counterparts. It has also been shown that unlike the kappa distribution where parallel electron beta was found to play the key role, whistler instability with Cairns distributed electrons shows a greater sensitivity towards electron temperature anisotropy. It has been shown that the real frequency of the whistler waves shows dependence on the choice of parallel electron beta. Interestingly, it has been found that a particular combination of parallel electron beta and electron temperature anisotropy is deleterious for the whistler instability. In the end, a comparison of whistler waves dispersion and growth has been done with elec tron temperature anisotropy using different non-thermal distributions by changing the plasma parameters. It has been found that the growth rate of whistler instability is maximum for flat topped distribution whereas it is minimum for the Maxwellian distribution. Our study may be beneficial to understand the propagation characteristics of waves in space plasmas where departure from Maxwellian behavior has been divulged by many expeditions in space.