In this report, we will introduce our recent developments in topological insulators and the corresponding topics.Firstly, we theoretically investigate the finite size effect in quantum anomalous Hall (QAH) system.Using Mn-doped HgTe quantum well as an example, we demonstrate that the coupling between the edge states is spin dependent and is related not only to the distance between the edges but also to the doping concentration.Thus with proper tuning of the two, we can get four kinds of transport regimes: quantum spin Hall, QAH, edge conducting, and normal insulator.These transport regimes have distinguishing edge conducting properties while the bulk is insulting.Our results give a general picture of the finite size effect in a QAH system, and are important for the transport experiments in QAH nanomaterials as well as future device applications [1].Secondly, we propose practical designs to realize topological field-effect quantum transistors in an HgTe nanoribbon.As a top gate is placed on the HgTe nanoribbon and with an increasing gate voltage, two new conductance channels develop in the HgTe nanoribbon.The quantum states in the new channels are not only robust against a short-range Anderson disorder, but can also couple with the intrinsic helical edge states in the boundaries of the HgTe nanoribbon to open a gap in the energy spectrum, indicating their topological characteristics.More importantly, the newly developed conductance channels can be turned on or off easily by adjusting the gate voltage.The proposal of controllable topological edge states produced by the gate voltage opens a new route for future topological field-effect quantum transistors in nanoelectronics and spintronics [2,3,4].