Understanding the homotopy groups of spheres is a major goal in homotopy theory. They have many applications such as the obstructions in surgery and the classification of differential structures on spheres. Currently, we only have a very limited knowledge of the structure of these groups. In this talk, I will survey the classical and new methods in the computations of stable and unstable homotopy groups, such as the Adams spectral sequence, the motivic methods and chromatic methods. These computations also show important structures in homotopy theory, such as the periodicity structure in stable and unstable homotopy groups, the finite exponent phenomenon in unstable homotopy.