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We study tetrahedron maps, which are set-theoretical solutions to theZamolodchikov tetrahedron equation, and Yang-Baxter maps, which areset-theoretical solutions to the quantum Yang-Baxter equation.  In particular, we clarify the structure of the nonlinear algebraic relationswhich define linear (parametric) tetrahedron maps (with nonlinear dependence onparameters), and we present several transformations which allow one to obtainnew such maps from known ones. Furthermore, we prove that the differential of a(nonlinear) tetrahedron map on a manifold is a tetrahedron map as well. Similarresults on the differentials of Yang-Baxter and entwining Yang-Baxter maps arealso presented.  Using the obtained general results, we construct new examples of (parametric)Yang-Baxter and tetrahedron maps. The considered examples include mapsassociated with integrable systems and matrix groups. In particular, we obtaina parametric family of new linear tetrahedron maps, which are linearapproximations for the nonlinear tetrahedron map constructed by Dimakis andM\"uller-Hoissen [arXiv:1708.05694] in a study of soliton solutions of vectorKadomtsev-Petviashvili (KP) equations. Also, we present invariants for thisnonlinear tetrahedron map.

Tetrahedron maps, Yang-Baxter maps, and partial linearisations