AD_ℝ is a natural strengthening of AD, which states that all games on real numbers are determined. Solovay proved that under AD_ℝ there is a canonical fine, normal countably complete measure on P_ω1(ℝ) (we will call such measures ℝ-supercompact). Moreover Woodin showed that the models of the form L(ℝ,μ) satisfying the theory "ZF+AD⁺ + μ is an ℝ-supercompact measure" satisfy as well "μ is the unique such measure". In recent work with Nam Trang, we proved that (modulo some large cardinals) the models of the form L(ℝ, μ) are unique (very much as Kunen's version of L[U]). I will give the outline of the mentioned results of Solovay, and Woodin, and discuss the proof of the uniqueness of such models.