The research presented here focuses on the study of phenomena associated with the sheaths present in capacitive rf discharges. A one-dimensional electrostatic current-driven computer simulation based on the Particle-In-Cell scheme is developed in order to treat a semi-infinite plasma in contact with an electrode. The plasma can be collisional or collisionless, ionisation is ignored and one or more ion species can be modelled. Primarily, the simulation is used to study the electron heating mechanism in the absence of collisions. The currently predominant theory is based on the idea of stochastic heating through a Fermi acceleration mechanism. Existing theoretical models dealing with the problem are sensitive to various questionable assumptions. The theory of stochastic heating based on Fermi acceleration is found to give zero net heating and therefore to be inadequate in providing an explanation for the source of heating in the collisionless case. In contrast, heating is shown to be the result of the alternating compression and rarefaction of the electron population by the oscillating sheaths. The conditions under which this mechanism gives a non-zero net result are described. An analytic fluid model is derived to illustrate the effect, and results obtained from this model show very good agreement with the simulation. This interpretation of collisionless heating illustrates the collective behaviour of the electrons, and also removes the role of the sheath edge as the source of collisionless heating. Finally the problem of the plasma-sheath transition through an intermediate “presheath” region is studied for the cases where one or two ion species are present. The presheath region characteristics are compared with existing theoretical models and good agreement is found. An investigation of the possible values of the average ion velocities compatible with the Bohm criterion, in terms of the values of the particle fluxes and the mean free paths, is performed for the multiple ion species case.