This dissertation undertakes the theory and methods of sufficient dimension reduction in the content of Hilbert-Schmidt Independence Criterion (HSIC). The proposed estimation methods enjoy model free property and require no link function to be smoothed or estimated. Two tests: Permutation test and Bootstrap test, are investigated to examine the true underlying dimension of data considered. Sampling distribution of our estimator is established in single-index regressions. Root-n consistency of our estimator is proved for multiple-index models. Finite sample performances are examined through simulation studies and compared with well-established methods, for example, SIR (Li, 1991), SAVE (Cook and Weisberg, 1991), rMAVE (Xia et al., 2002), and EFM (Cui et al., 2011). Two real data sets are analyzed to demonstrate the efficacy of our proposed approaches. We also explored the use of HSIC for feature screening purpose. Our procedures do not require model specification for responses and predictors, for example, linear model or generalized linear model, discrete or non-normal predictors, which makes our methods robust against model misspecification. In addition, differing from usual screening approaches in which only marginal relations are used, our procedure provides a sufficient selection path by using joint relations among the response variable and predictors.