The quantum Hall effect refers to the quantization of charge Hall resistance (in units of h/e2, where h is the Planck constant and e is the electronic charge), observed in a quantum well when an intense magnetic field is applied at low temperatures. The Quantum Spin Hall Effect (QSHE) is an emergent phenomenon recently discovered in semiconductors and other novel materials, where the intrinsic spin Hall conductance is quantized in the absence of any magnetic field. This talk will report on our experimental efforts in low temperature quantum transport of InAs/GaAs quantum wells, in which an inverted band structure can be engineered and fine-tuned by electrostatic gates. We describe experimental results showing energy gap and edge transport in this system. The prospects for observing QSHE in this material as suggested by recent theories, will be discussed.