Curse of dimensionality refers to sparse phenomena of high-dimensional data and associated challenges in statistical analysis. Traditional nonparametric methods provide flexible modeling tools to discover nonlinear and complex patterns in data, but they often experience theoretical and computational difficulties when handling high-dimensional data. Over the past two decades, rapid advances have occurred in nonparametrics to break the curse of dimensionality. A variety of state-of-art nonparametric methods, theory, and scalable algorithms have been developed to extract low intrinsic dimension from data and accommodate high-dimensional data analysis more effectively. In this talk, I will survey some recent works of nonparametric methods in model estimation, variable selection, and inferences for high dimensional regression, classification, and density estimation problems. Related issues and open challenges will be discussed as well.