In this thesis, we have studied the propagation of nonlinear electrostatic waves in a nonMaxwellian plasma in which electrons follow the generalized (??,??) distribution function which has the advantage that it mimics most of the distribution functions observed in space plasmas. Generally spectral index ?? corresponds to the flatness and ?? corresponds to the high energy tail in the profile of the distribution. The generalized (??,??) distribution function is the generalized form of kappa and Maxwellian distribution function and can be reduced to these in the limits ??=0,??=(??+1) and ??=0,??→∞, respectively. First we studied the propagation of nonlinear ion acoustic waves in a (??,??) distributed plasma and derived the Korteweg-de Vries (KdV) equation. In the past, KdV equation has been derived for ion acoustic waves with Boltzmannian or kappa distributed electrons and only compressive solitary structures were obtained. We have shown that when electrons are modelled by (??,??) distribution, the nonlinear ion acoustic waves admit both humps and dips in the perturbed potential. We found that for positive values of ??, which correspond to flat-topped distribution, the nonlinear ion acoustic waves admit compressive solitary structures or density humps. And for negative values of ??, which correspond to a spiky distribution, the nonlinear ion acoustic waves admit rarefactive solitary structures or density depletions. It has also been shown that the generalized (??,??) distribution function provides another way to explicate the density depletions observed by Freja and Viking satellites previously explained by proposing Cairns distribution function. In the third chapter, we have studied the propagation of nonlinear electron acoustic waves (EAWs) by deriving the KdV equation in a plasma comprising of cold and hot electron populations in which the ions form the neutralizing background. The hot electrons have been assumed to follow the generalized (??,??) distribution. Interestingly, it has been found that unlike Maxwellian and kappa distributions, the electron acoustic waves admit not only rarefactive structures but also allow the formation of compressive solitary structures for generalized (??,??) distribution. Using the plasma parameters, typically found in the Saturn’s magnetosphere and the Earth’s auroral region, where two x populations of electrons and electron acoustic solitary waves have been observed, we have given an estimate of the scale lengths over which these nonlinear waves are expected to form and how the size of these structures would vary with the change in the shape of the distribution function and with the change of the plasma parameters. In chapter-4, we then derived the modified Korteweg-de Vries (mKdV) equation to study nonlinear ion acoustic waves in a plasma in which electrons follow generalized (??,??) distribution. The spectral index ?? in the distribution corresponds to the flat top at low energy and by increasing its value flat top in the distribution increases. The spectral index ?? can also have negative values due to which distribution becomes spiky at low energies. Such flat topped or spiky distributions have been frequently observed in space plasmas. By employing (??,??) distribution, it has been shown that solitary structures are much influenced by the spectral index ??. This study highlighted the effect of low energy particles on the propagation characteristics of the solitary structures which could not be done by employing Maxwellian or kappa distributions and be helpful in explaining the underline physics in those regions where such flat top distributions are observed. In the chapter 5, we have studied the propagation of nonlinear ion acoustic shock waves in unmagnetized and collisionless plasma in the presence of electrons that follow the generalized (??,??) distribution. The Burger and Korteweg-de Vries–Burger (KdV-Burger) equations have been derived through reductive perturbation technique and via tangent hyperbolic method shock like solutions have been presented analytically for both Burger and KdV-Burger equations. It has been found that the strength and steepness of shock waves are affected by the flatness parameter ??, tail parameter ?? and the nonlinear propagation velocity ??. It is found that the shock strength of KdV–Burger is less than simple Burger equation’s shock. We concluded that propagation of nonlinear electrostatic waves strongly dependent on the profile of the distribution function and the results obtained are of great importance as they interpret those observations which could not be predicted on the basis of Maxwellian or kappa distribution functions.