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Thouless pump with quantized transports is topologically robust against small perturbations and disorders, while breaks down under sufficiently strong disorders. Here we propose counter-intuitive topological pumps induced by disorders in noninteracting and interacting systems. We first show an extrinsic topological pump driven by the on-site quasiperiodic potential for a two-loop sequence, where the disorder inequivalently suppresses the topology of two pump loops. Moreover, we reveal an intrinsic topological pump induced by the hopping quasiperiodic disorder from a trivial single-loop pump in the clean limit, dubbed the topological Anderson-Thouless pump (TATP) as a dynamical analogue of topological Anderson insulators. We demonstrate that the mechanism of the TATP is the disorder-induced shift of gapless critical points and the TATP can even exhibit in the dynamic disorder and interacting cases. Finally, we extend the TATP to higher-order topological systems with disorder-induced quantized corner transports. Our proposed TATPs present new members of the topological pump family and could be realized with ultracold atoms or photonic waveguides.