本文探究了 Wasserstein 距离和 Gromov-Wasserstein 距离的本质。先是简要叙述了 Wasserstein 距离的基本性质和概率表达形式，并以离散型的概率分布以及高斯分布为例来进行具体的计算。然后研究了其他概率度量Total Variation、KL 散度、JS 散度、Hellinger 距离的性质和其推广形式，以及各自在统计距离中基于函数 f 的 f-散度形式，并且通过对比凸显Wasserstein 距离作为度量的优点。紧接着对 Gromov-Wasserstein 距离的性质进行研究，给出其应用举例，并与 Wasserstein 距离进行比较。最后介绍了非平衡最优运输下的 Wasserstein 距离和 Gromov-Wasserstein 距离。

This paper explores the nature of Wasserstein distance and Gromov-Wasserstein distance. First, the basic properties and probability expression of Wasserstein distance are briefly described.Then the discrete probability distribution and Gaussian distribution are taken as examples for specific calculation. Then, the properties and generalized forms of other probability measures, such as Total Variation, KL divergence, JS divergence and Hellinger distance, as well as their respective f-divergence forms based on function f in statistical distance, are studied, and the advantages of Wasserstein distance as a metric are highlighted by comparison.Then the properties of Gromov-Wasserstein distance are studied, its application examples are given, and the comparison with Wasserstein distance is made. Finally, we introduce Wasserstein distance and Gromov-Wasserstein distance under unbalanced optimal transport.