We have developed a scaling theory of polyelectrolyte adsorption at an oppositely charged surface. At low surface charge densities, we predict two-dimensional adsorbed layers with thickness determined by the balance between electrostatic attraction to the charged surface and chain entropy. At high surface charge densities, we expect a 3-dimensional layer with a density profile determined by the balance between electrostatic attraction and short-range monomer-monomer repulsion. These different stabilizing mechanisms result in a nonmonotonic dependence of the layer thickness on the surface charge density. For adsorption of polyelectrolyte chains from salt solutions, the screening of the electrostatic repulsion between adsorbed polyelectrolyte chains results in large overcompensation of the surface charge for two-dimensional adsorbed layers. At higher salt concentrations this overcompensation of the surface charge by the 2-d adsorbed layer is independent of the original surface charge and depends only on the fraction of the charged monomers on the polyelectrolyte chains and increases with ionic strength. The polyelectrolyte surface excess in 3-d adsorbed layers increases at low ionic strength and decreases at higher ionic strength.

In dilute solutions of rodlike polyelectrolytes some counterions are distributed far from polyions while others are located in their vicinity in the regions of cylindrical symmetry of the electrostatic potential. For these cylindrical regions around rodlike polyelectrolytes we find an exact solution of the nonlinear Poisson-Boltzmann equation for the case of nonzero net charge in these regions. This exact solution implies three qualitatively different phases of counterion distribution around the polyions with second order phase transitions between these phases.