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Andersons groundbreaking discovery that the presence of stochastic imperfections in a crystal may result in a sudden breakdown of conductivity revolutionized our understanding of disordered media. After stimulating decades of lively studies, Anderson localization has found intriguing applications in various areas of physics, such mesoscopic physics, strongly-correlated systems, light localization, cavity quantum electrodynamics, random lasers, and topological phases of matter. However, a fundamental assumption in Andersons treatment is that no energy is exchanged with the environment, in contrast to the common knowledge that every real system is subject to dissipation. Recently, a growing number of theoretical studies has addressed disordered media with dissipation. In particular it has been predicted that in such systems all eigenstates exponentially localize, similar to the original case without dissipation that Anderson considered. However, in dissipative systems an eigenstate analysis is insufficient for characterizing the transport dynamics of wave, in stark contrast to Hermitian systems, where the localization of all eigenstates necessarily suppresses transport. In our work, we show in theory and experiment that systems with dissipative disorder allow for a new type of spatial transport, despite the fact that all eigenstates are exponentially localized. This Anderson transport is characterized by super-diffusive spreading and ultra-far spatial jumps between localized states. We anticipate our findings to mark the starting point towards novel phenomena in dissipative media, which are subject to general non-Hermitian disorder.