This doctoral dissertation has considered several models to study the nonlinear dynamics of ion-acoustic electrostatic solitary and shock waves in a variety of non-Maxwellian plasmas using a fluid equation approach. The properties of ion acoustic solitary waves in both magnetised and unmagnetised three-dimensional collisionless plasmas containing excess superthermal electrons are investigated from first principles. With higher superthermality, the phase speed and dispersion of the waves are shown to be suppressed, and nonlinearity increases, leading to structures with smaller amplitudes and narrower widths. In magnetised plasmas the solitary wave width decreases with increased magnetic field strength. When dissipation is taken into account, using the fluid kinematic viscosity, electrostatic shock solutions can be found. Shocks become larger and narrower with stronger magnetic fields, higher superthermality and higher viscosity. The role of positrons is also explored. Both conservative and dissipative models are reviewed, in magnetised and unmagnetised plasma. Solitary wave frequency and phase speed decreases and structures become smaller and narrower when the positron concentration increases, and they increase in amplitude and width if the electron/positron temperature ratio is increased. Shock excitation amplitudes increase with increasing positron concentration, while widths decrease. A hybrid distribution function is employed combining a Cairns-type nonthermal form with the Tsallis theory for nonextensive thermodynamics. Arbitrary amplitude ion acoustic solitary wave dynamics in a two-component plasma are investigated, and the valid range of the function examined. Solitary waves are shown to exist within a narrow range of allowable Mach numbers. Both positive and negative potential structures are found, and coexistence may occur. Finally an unmagnetised collisionless electron-ion plasma model is proposed, featuring a non-Maxwellian-trapped electron distribution, modelled by a kappa distribution function combined with a Schamel distribution. With enhanced superthermality, the amplitude and width of solitary waves decreases. Shock waves are shown to be possible and their dynamics discussed.