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Motivated by the task of sampling measures in high dimensions we will discuss a several gradient flows in the spaces of measures, including the Wasserstein gradient flows of Maximum Mean Discrepancy and relative entropy, the Stein Variational Gradient Descent and a new Radon-Wasserstein gradient flows.   For all the flows we will consider their deterministic interacting-particle approximations. The talk will highlight some of the properties of the flows and indicate their differences. In particular we will discuss how well can the interacting particles approximate the target measures. The talk is based on joint works with Elias Hess-Childs, Anna Korba, Sangmin Park,  Lihan Wang, and Lantian Xu.