The quantum spin Hall is a newly conjectured state of matter that does not break any discrete symmetries such as time reversal or parity. The state requires spin-orbit coupling and is protected by a Z₂ topological invariant. A new class of materials called topological insulators are able to support spin transport both in two and three dimensions through stable edge and/or surface states.  We predict that a non-trivial topological band insulator that gives rise to a Quantum spin Hall state must exist in a series of quantum wells of the inverted-band-structure variety such as HgTe. A k=0, the inverted band structure in HgTe/CdTe quantum wells gives rise to a single Dirac Fermion without the doubling problem, and it can be proved that the quantum spin Hall state exists in a range of quantum well thicknesses.  The interplay of a non-trivial topology of the momentum space in the anomalous Quantum Hall effect and the real-space topology in the Quantum Hall effect arising from an external magnetic field is also analyzed in a generalized transfer-matrix formalism.