  The von Neumann lattice representation is a convenient representation for
studying several intriguing physics of quantum Hall systems. In this formalism,
electrons are mapped to lattice fermions. A topological invariant expression of
the Hall conductance is derived and is used for the proof of the integer
quantum Hall effect in the realistic situation. Anisotropic quantum Hall gas is
investigated based on the Hartree-Fock approximation in the same formalism.
Thermodynamic properties, transport properties, and unusual response under
external modulations are found. Implications for the integer quantum Hall
effect in the finite systems are also studied and a new quantum Hall regime
with non-zero longitudinal resistance is shown to exist.
