Abstract:In the talk we will review compactness results and regularity theorems related to harmonic maps. We will review Sacks-Uhlenbeck blow-up methods, Hélein regularity theorem, Evens and Bethuel partial regularity, Lin theorem, Rivière estimates, and our results in this field. Especially, we will talk approximated harmonic maps from surfaces with tension field bounded in Zygmund class, the blow up formulafor stationary harmonic maps, and it’s applications to triholomorphic maps between Hyperkähler manifolds.