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Clustering on uncertain data, one of the essential tasks in mining uncertain data, posts significant challenges on both modeling similarity between uncertain objects and developing efficient computational methods. The previous methods extend traditional partitioning clustering methods like $(k)$-means and density-based clustering methods like DBSCAN to uncertain data, thus rely on geometric distances between objects. Such methods cannot handle uncertain objects that are geometrically indistinguishable, such as products with the same mean but very different variances in customer ratings. Surprisingly, probability distributions, which are essential characteristics of uncertain objects, have not been considered in measuring similarity between uncertain objects. In this paper, we systematically model uncertain objects in both continuous and discrete domains, where an uncertain object is modeled as a continuous and discrete random variable, respectively. We use the well-known Kullback-Leibler divergence to measure similarity between uncertain objects in both the continuous and discrete cases, and integrate it into partitioning and density-based clustering methods to cluster uncertain objects. Nevertheless, a naïve implementation is very costly. Particularly, computing exact KL divergence in the continuous case is very costly or even infeasible. To tackle the problem, we estimate KL divergence in the continuous case by kernel density estimation and employ the fast Gauss transform technique to further speed up the computation. Our extensive experiment results verify the effectiveness, efficiency, and scalability of our approaches.