Indifferent fixed points of holomorphic maps give rise to delicate problems such as linearization problems, discontinuous Julia sets etc. In this talk, we review the study of those phenomena from the renormalization point of view. We define a certain class of holomorphic maps with "non-degenerate" parabolic fixed points. The parabolic renormalizaion is defined for this class and shown to leave the class invariant. Then the class is still invariant under a small perturbation, therefore we can handle irrationally indifferent fixed points with large continued fraction coefficients. This is a joint work with Hiroyuki Inou.

We will compare this approach with Yoccoz's and McMullen's renormalizations. We will also mention possible applications such as Buff-Cheritat's work toward positive measure Julia sets.