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In this paper, a distributed Student's t filtering algorithm to deal with heavy-tailed noises is developed. In the traditional Kalman filter, the distribution of the signal is assumed. However, in reality, outliers in the signal are often encountered for which the assumption of Gaussian distribution is no longer valid. The Student's t distribution can describe noises in the presence of outliers.As a result, the weight on each data point within the filter adapts to the data quality so that the filter becomes insensitive to the outliers. We first derive the distributed filtering algorithm from the centralized Student's t filter, which is able to handle heavy-tailed noises such as outliers and then analyze properties of the proposed method. It is shown that the proposed algorithm provides the same accuracy as the centralized Student's t filtering with no performance loss. Furthermore, the distributed Student's t filtering with feedback is developed, which is in accordance with centralized filtering, and the local error covariance is reduced as expected. Two numerical examples support the theoretical results and illustrate the validity of the proposed method.