In this tutorial and review paper, we investigate the influence of trapped particles on the propagation of ion acoustic solitons and the role of trapped waves on the propagation of Langmuir solitons. The classical potential method allows us to construct finite amplitude soliton solutions, and trapping phenomena are found to retard the motion of solitons. Thus, ion acoustic solitons have minimum speed when they are based on an isothermal electron equation of state since this corresponds to "maximum electron trapping". In the small amplitude regime, deviations from the Boltzmann law lead to a new nonlinear term. Subsequently, the dynamics of the soliton is governed by a modified KdV-equation [e.g., (23)]. For Langmuir solitons an existence diagram is found and exhibits the comparison between experimentally observed localized structures and theory. The connection with the various small amplitude soliton solutions is also pointed out. Moreover, the inclusion of a pump and of dissipative terms in the coupled nonlinear Schrödinger-ion equation gives rise to a transient phenomenon called soliton flash, whose implication to the laboratory experiments is discussed. Finally, as an application in numerical analysis the propagation of solitons is followed by solving the KdV-equation in Fourier-space. This turns out to be an excellent tool to test stability and accuracy of numerical schemes. Our results reveal that the so-called aliasing interactions lead to erroneous solutions. The conservation of invariants does not guarantee the accuracy of a numerical algorithm, although it is needed for its stability.