In low pressure plasmas the transition between the prlsheirth and the sheath is investigated taking into account solutions o f a two-fluid model in the whole plasma. The kinetic theory is reviewed only shortly. The hydrodynamic theory is described in detail. In addition, new results are given. Boundary conditions necessary f o r the hydrodynamic description are derived from the electron kinitics. The two-fluid equations supplemented by the Poisson equation constitute a set o f nonlinear differential equations with isoiated singular pointr. In addition, this is a singularly perturbated problem if the space charge sheath is thin. Singular points occur if the ions or the electrons attain the corresponding sound speed. The mathematical probiem can be split into a boundary value problem f o r subsonic ion velocities and in an initial value problem ?or the hypersonic range. Solutions are obtained by combination o f analytical and numerical methods. The numerical integration involvcsinstabilities in the subsonic range if simultaneously the plasma is collision dominated and the space charge density is very small in the interior. Several methods are given to reduce o r to avoid these difficulties. It is shown that a magnetic field, directed parallel to the tube wall, can reduce the voltage drop across the sheath substantially. In a positive column the radial variation of the electron temperature is calculated by means o f the twofluid model. Sheath criteria are discussed. For simplification a plane and one-dimensional model is* considered. In Cartesian coordinates the walls are taken at x=+d. Steady-state conditions and homogeneity parallel to the walls are assumed to prevail throughout the plasma. Ions and electrons are generated within the plasma, flow to the walls, recombine there and re-enter the plasma am neutral atoms.,The insulating walls are charged negatively with respect to the plasma. The arising electric field equalises the fluxes of electrons and ions to the walls. This model describes features o f the