An analysis of waves and instabilities in pair-ion plasma produced in some recent experiments is presented. The pair-ions, C±60, have the same mass and opposite charge. It is pointed out that the observation of the electrostatic ion acoustic wave frequency can be a suitable check to determine whether the produced plasma is a pure pair-ion plasma or whether it comprises some concentration of electrons. Linear and nonlinear electrostatic waves are studied in an unmagnetized pure pair-ion (PI) and pair-ion–electron (PIE) plasmas. In this thesis, we primarily focus on the analytical and numerical study of linear and nonlinear waves in pair-ion and pair-ion-electron plasmas and in particular formation of three different types of nonlinear structures have been investigated. The shear flow-driven electrostatic instabilities has been investigated in ideal low-density, low-temperature pair-ion-electron and pure pair-ion plasmas in several different cases, by considering homogeneous and inhomogeneous density effects. In uniform pair-ionelectron plasma, when the shear flow is of the order of the acoustic speed, a purely growing D''Angelo mode can give rise to electrostatic fields. In the case of an inhomogeneous plasma, the drift wave becomes unstable. The presence of negative ions, however, reduces the growth rate. If the positive and negative ions are not in thermal equilibrium with each other, then the shear flow also gives rise to an electrostatic instability in pure pair-ion plasma. The Kortewege-de Vries-Burgers (KdV-B) equation is derived for drift-waves in a partially ionized non-uniform pair-ion-electron (PIE) plasma. The nonlinearity appears due to electron temperature gradient. The analytical solutions in the form of solitons, monotonic shocks and oscillatory shocks have been obtained. The numerical calculations have also been presented for PIE plasmas of fullerene and hydrogen for illustration keeping in view the recent experiments. The Kadomtsev-Petviashvili-Burgers (KPB) equation is also derived for coupled drift acoustic shock waves in a partially ionized non-uniform pair-ion-electron (PIE) plasma in the presence of both density and temperature gradients, respectively. Both linear and nonlinear studies have been presented. The nonlinear KPB equation is derived in the small amplitude approximation method and its solution is found using the tanh method. The numerical calculations have also been presented for PIE plasmas of fullerene plasma. The effect of density and temperature inhomogeneities on the nature of the shock is also highlighted. The role of the velocity of the nonlinear structure with regard to the density and temperature gradients driven drift velocities is also pointed out and the effect of ionneutral collision frequency is also investigated. The linear and nonlinear dynamics of pair-ion (PI) and pair-ion-electron plasmas (PIE) have been investigated in a cylindrical geometry with a sheared plasma flow along the axial direction having radial dependence. The coupled linear dispersion relation of lowfrequency electrostatic waves has been presented taking into account the Gaussian profile of density and linear gradient of sheared flow. It is pointed out that the quasi-neutral cold inhomogeneous pure pair ion plasma supports only the obliquely propagating convective cell mode. The linear dispersion relation of this mode has been solved using boundary conditions. The nonlinear structures in the form of vortices formed by different waves have been discussed for PI and PIE plasmas. Johnson''s equation which is also known as cylindrical Kadomstev-Petviashvili (CKP) equation, is derived for pair-ion-electron plasmas to study the propagation and interaction of two solitons. Using a novel gauge transformation, two soliton solutions of CKP equation are analytically solved by using the Hirota''s method. Interestingly, it is observed that unlike the planar Kadomstev-Petviashvili (KP) equation, the CKP equation admits horseshoe-like solitary structures. Another non-trivial feature of CKP solitary solution is that the interaction parameter gets modified by the plasma parameters on contrary to the one obtained for Korteweg de Vries (KdV) type equation.