This article provides a comprehensive overview of variable selection in high-dimensional feature spaces. It discusses the challenges and recent developments in statistical methods for high-dimensional data analysis, emphasizing the role of penalized likelihood and its statistical properties. The paper highlights the importance of variable selection in improving estimation accuracy, model interpretability, and computational efficiency in high-dimensional statistical modeling. It reviews various penalized likelihood methods, including LASSO, SCAD, and MCP, and discusses their properties, such as sparsity, unbiasedness, and continuity. The article also addresses the computational challenges of high-dimensional data and proposes methods like sure screening and two-scale approaches to handle ultra-high dimensional variable selection. It emphasizes the need for robust and efficient algorithms to select important variables and estimate their effects in high-dimensional statistical inference. The paper concludes with a discussion of the importance of variable selection in statistical learning and scientific discoveries, and the ongoing research in this area.This article provides a comprehensive overview of variable selection in high-dimensional feature spaces. It discusses the challenges and recent developments in statistical methods for high-dimensional data analysis, emphasizing the role of penalized likelihood and its statistical properties. The paper highlights the importance of variable selection in improving estimation accuracy, model interpretability, and computational efficiency in high-dimensional statistical modeling. It reviews various penalized likelihood methods, including LASSO, SCAD, and MCP, and discusses their properties, such as sparsity, unbiasedness, and continuity. The article also addresses the computational challenges of high-dimensional data and proposes methods like sure screening and two-scale approaches to handle ultra-high dimensional variable selection. It emphasizes the need for robust and efficient algorithms to select important variables and estimate their effects in high-dimensional statistical inference. The paper concludes with a discussion of the importance of variable selection in statistical learning and scientific discoveries, and the ongoing research in this area.