High dimensional statistical problems arise from various fields of scientific research and technological development. Variable selection is a fundamental issue for high-dimensional statistical modeling. The traditional idea of best subset selection methods is computationally expensive and unstable. As an alternative, many penalized methods have been successfully developed over the last decade. The representable examples are the least absolute shrinkage and selection operator (LASSO) and smoothly clipped absolute deviation (SCAD). They have been widely applied to various high-dimensional statistical problems. In this talk, we introduce various penalized methods and the related issues: concepts, theoretical backgrounds and computational algorithms. In addition, we show their various applications to grouped or structured variable selection and Gaussian network problems.