Differential equations can be solved in only limited cases. So it is worth to study Liouville integrability for Hamiltonian system. Meanwhile, differential Galois theory for linear ordinary differential equations has developed by Picard-Vessiot and it also gives the information of solutions. In recent years, Morales-Ramis theory which associate these two different point of view and its extension by Ayoul-Zung provide a lot of results about integrability. Then, in this presentation, these method and their algebraic background is introduced. Especially, the outline of Picard-Vessiot theory is presented with a comparison to classical Galois theory for algebraic equations.