In this thesis, for a locally compact quantum grou , we initiate a study of topological inner amenability for . We show that all amenable or co-amenable locally compact quantum groups are topologically inner amenable. We then show that topological inner amenability of is equivalent to the existence of certain functionals associated to characters on . For co-amenable locally compact quantum groups, we introduce and study strict topological inner amenability and its relation to extensions of the co-unit from to . We then obtain a number of equivalent statements describing strict topological inner amenability of and the existence of certain means on suaces of uch as , and . Finally, we introduce the concepts of weak* and weak topological inner fixed point. Then we study the relationship between topological inner amenability of and the existence of weak and weak* topological inner fixed points and the common fixed points.