The paper discusses the importance of choosing appropriate probability metrics when studying the convergence of measures. It provides a comprehensive review of ten commonly used probability metrics, including the discrepancy, Hellinger distance, relative entropy, Kolmogorov metric, Lévy metric, Prokhorov metric, separation distance, total variation distance, Wasserstein metric, and $\chi^2$-distance. The authors present new bounds and relationships among these metrics, highlighting how the choice of metric can significantly affect the rates and nature of convergence. They also illustrate these relationships through examples and applications, emphasizing the practical implications of these metrics in various statistical and probabilistic contexts. The paper concludes with a discussion on the importance of carefully selecting a metric to ensure accurate and meaningful convergence analysis.The paper discusses the importance of choosing appropriate probability metrics when studying the convergence of measures. It provides a comprehensive review of ten commonly used probability metrics, including the discrepancy, Hellinger distance, relative entropy, Kolmogorov metric, Lévy metric, Prokhorov metric, separation distance, total variation distance, Wasserstein metric, and $\chi^2$-distance. The authors present new bounds and relationships among these metrics, highlighting how the choice of metric can significantly affect the rates and nature of convergence. They also illustrate these relationships through examples and applications, emphasizing the practical implications of these metrics in various statistical and probabilistic contexts. The paper concludes with a discussion on the importance of carefully selecting a metric to ensure accurate and meaningful convergence analysis.