The investigation of perpendicularly propagating modes, excitations and associ- ated instabilities at the order of electron cyclotron frequencies in non-relativistic plas- mas is important for understanding many astrophysical and laboratory phenomenon such as noise emission and absorption, solar wind plasma, large number of experiments lunched with satellites to provide in situ data on the properties and nature of plasmas in the earth and other planetary magnetospheres, the achievement of thermonuclear fusion and so forth. In this context, the Vlasov model is employed for electron-ion plasma in which the ion dynamics are ignored. Using kinetic theory approach for homogenous collisionless magnetized plasma, we derive the general expression for the conductivity tensor in cylindrical polar coordinates. Modes of non-relativistic electrons are investigated for perpendicular propagation in non-Maxwellian plasma. For this purpose thermal ring and drifting Maxwellian distribution functions are used to derive di¤erent modes. In this thesis we particularly focus on analytical and numerical solution of the dis- persion relation for electrostatic Bernstein wave and electromagnetic O mode which propagate perpendicular to the ambient magneticeld. The classic Bernstein waves may be intimately related to banded emissions detected in laboratory plasmas, terres- trial and other planetary magnetospheres. However, the customary discussion is based upon isotropic thermal velocity distribution function. In order to understand how such waves are excited one needs an emission mechanism, i.e., instability. In non-relativistic collision-less plasmas, the only known Bernstein wave instability is that associated with a cold perpendicular velocity ring distribution function. However, cold ring distribu- tion is highly idealized. The present thesis generalizes the cold ring distribution to include thermal spread, so that the Bernstein-ring instability is described by a more realistic electron distribution function, with which the stabilization by thermal spread associated with the ring distribution is demonstrated. The presentndings imply that the excitation of Bernstein waves requires a su¢ ciently high perpendicular velocity gradient associated with the electron distribution function. The O mode is unstable against temperature anisotropic plasma having Tk > T? (where k and ? corresponds to the direction with respect to external magneticeld B0). These purely growing waves has great importance due to its possible application to the solar wind plasma. In past huge amount of literature on O mode instability has been devoted to either bi-Maxwellian or counterstreaming velocity distribution. For solar wind plasma trapped in a magnetic mirror-like geometry for instance magnetic clouds or in the locality of the Earth s collisionless bow shock environments, the velocity distribution function may hold a loss-cone feature. In situations like these the O mode instability may be excited for cyclotron harmonics as well as the purely-growing branch. We investigates the Omode instability for plasmas characterized by the parallel Maxwellian distribution and perpendicular thermal ring velocity distribution in order to understand the general stability characteristics of the electromagnetic O mode. The purely growing ordinary O mode instability wasrst discussed by Davidson and Wu [Phys. Fluids 13, 1407 (1970)]. In a series of papers, Ibscher, Schlickeiser, and their colleagues [Phys. Plasmas 19, 072116 (2012); ibid. 20, 012103 (2013); ibid. 20, 042121 (2013); ibid. 21, 022110 (2014)] revisited the O mode instability and extended its application to the low-beta plasma regime by considering a counter-streaming bi- Maxwellian model. However, the O mode instability is thus far discussed only on the basis of the marginal stability condition rather than actual numerical solutions of the dispersion relation. In the present thesis we re-examin the O mode instability by considering the actual complex roots. The marginal stability condition as a function of the (electron) temperature anisotropy and beta naturally emerges in such a scheme.