This paper provides a comprehensive introduction to the Complex Structured Singular Value ($\mu$), focusing on its mathematical aspects. $\mu$ is a key tool for analyzing the robustness and performance of linear feedback systems. The paper covers the definition and properties of $\mu$, including bounds and computational methods. It discusses the relationship between $\mu$ and Linear Fractional Transformations (LFTs), which are essential for representing uncertain systems. The Main Loop Theorem is introduced, which links the robustness of linear fractional transformations to $\mu$. The paper also explores robustness tests using $\mu$, including stability and performance margins. Additionally, it presents a maximum modulus theorem for LFTs, which generalizes the classical maximum modulus theorem for rational functions. The paper concludes with a discussion on related work and future directions.This paper provides a comprehensive introduction to the Complex Structured Singular Value ($\mu$), focusing on its mathematical aspects. $\mu$ is a key tool for analyzing the robustness and performance of linear feedback systems. The paper covers the definition and properties of $\mu$, including bounds and computational methods. It discusses the relationship between $\mu$ and Linear Fractional Transformations (LFTs), which are essential for representing uncertain systems. The Main Loop Theorem is introduced, which links the robustness of linear fractional transformations to $\mu$. The paper also explores robustness tests using $\mu$, including stability and performance margins. Additionally, it presents a maximum modulus theorem for LFTs, which generalizes the classical maximum modulus theorem for rational functions. The paper concludes with a discussion on related work and future directions.