Quantum unique ergodicity conjecture discusses the limiting behavior of eigenfunctions of Laplacian on compact negatively curved manifolds. Results so far have connected the research areas of number theory, spectral theory and ergodic theory. In this talk, a general introduction to quantum unique ergodicity conjecture will be given, including current results, limitations of current methods, and how to get into this research field. Then the construction of arithmetic Fuchsian group will be introduced, which is the case considered in Elon Lindenstrauss’s paper: Invariant measures and arithmetic quantum unique ergodicity, Annals of Mathematics, 2006.