Two-dimensional electron systems, which exist e.g. at interlaces between two different semiconductors, exhibit interesting physical properties under strong magnetic fields. In interpreting the quantum Hall effect the role of one-dimensional system edges begins to be taken into account. The electron structure, connected with Landau quantization of 2D electron states under magnetic field, has been studied in the vicinity of system edges. Model systems with abrupt confinement barriers exhibit electron dispersions with edge plateaus above the barrier tops, accompanied by regions of substantially reduced gaps between neighbouring Landau branches. Selfconsistent results for smoothly confined systems provide alternating channels of compressible and incompressible Fermi liquids along the system edges. Recent investigation illustrates the transition between the two limiting confinement barrier cases. In order to evaluate the Hall conductivity, the Kubo formula has been adopted in a straightforward manner to two-dimensional stripes confined by arbitrary barriers. Total deviation of the Hall conductivity from the integer values is given by the product of two factors: the geometrical factor is inversely proportional to the sample width and the edge factor is proportional to the derivative of the electron dispersion at the Fermi level and is thus governed by the shape of the confinement barrier. The deviations have been evaluated for model systems of various widths and a qualitative agreement with recent experimental data for quantum wires has been found. The formulas provide also current densities and this enables to investigate spatial distributions of the electron current across the Hall stripes. Application to the abruptly confined model shows that the quantized part of the total current takes place within the interior of the stripe whereas the edge currents distribution is affected by the confinement barrier.