The Steinhaus Tiling problem (can one find a subset of the plane which tiles the plane when translated by the integer lattice, no matter how one rotates that subset?) has several variants, some of which are solved and some not. I will review the problem and progress on it during the last 30 years and will also speak about two variants that were resolved recently and whose resolution involves some independently interesting statements in harmonic analysis and geometry. Joint work with Michael Papadimitrakis.