Quantum Hall states are prototypical topological states of matter whose Hall conductance is topologically quantized to an integer or rational fraction multiple of the fundamental conductance quantum. A significant consequence of this quantization is that the Hall conductance value can be made independent of variations from device to device, within acceptable limits. Such topologically quantized properties are thus highly desirable for metrology or industrial purposes. Formulating a microscopic picture of fractional quantum Hall states and the characterization of all topological responses of quantum Hall states are frontier areas of condensed matter research, with far reaching technological consequences such as realizing anyonic topological quantum computation. In this dissertation, I will present my research on these topics.<br>