We give a brief survey of recent results on word maps on simple groups and polynomial maps on simple associative and Lie algebras. Our focus is on parallelism between these theories, allowing one to state many new open problems and giving new ways for solving older ones.
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ACKNOWLEDGMENTS This research was supported by the Israel Science Foundation, grant 1207/12. Kunyavskii and Plotkin were supported in part by the Minerva Foundation through the Emmy Noether Research Institute of Mathematics. Some questions raised in this paper were formulated after discussions with N.L. Gordeev, L.H. Rowen, A. Shalev, N.A. Vavilov, and E.I. Zelmanov to whom we are heartily grateful.