Topological full groups, as an algebraic invariant, were introduced to study continuous orbit equivalence relations by Giordano, Putnam, and Skau. Then, these groups have been found applications to geometric group theory by providing interesting examples with certain properties such as simplicity, soficity, amenability, and LEF-ness. In this talk, I will show new methods of establishing the soficity and LEF-ness for topological full groups. Moreover, I will explain how can one obtain amenability from the sofic approximations when the acting group is amenable and the action is distal.