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Hybrid AC/DC distribution systems are becoming a popular means to accommodate the increasing penetration of distributed energy resources and flexible loads. This paper proposes a distributed and robust state estimation (DRSE) method for hybrid AC/DC distribution systems using multiple sources of data. In the proposed distributed implementation framework, a unified robust linear state estimation model is derived for each AC and DC regions, where the regions are connected via AC/DC converters and only limited information exchange is needed. To enhance the estimation accuracy of the areas with low measurement coverage, a deep neural network (DNN) is used to extract hidden system statistical information and allow deriving nodal power injections that keep up with the real-time measurement update rate. This provides the way of integrating smart meter data, SCADA measurements and zero injections together for state estimation. Simulations on two hybrid AC/DC distribution systems show that the proposed DRSE has only slight accuracy loss by the linearization formulation but offers robustness of suppressing bad data automatically, as well as benefits of improving computational efficiency.