A model of the Axiom of Determinacy in which every set of reals is universally Baire

Abstract The consistency of the theory $\mathsf {ZF} + \mathsf {AD}_{\mathbb {R}} + {}$ ‘every set of reals is universally Baire’ is proved relative to $\mathsf {ZFC} + {}$ ‘there is a cardinal that is a limit of Woodin cardinals and of strong cardinals’. The proof is based on the derived model construction, which was used by Woodin to show that the theory $\mathsf {ZF} + \mathsf {AD}_{\mathbb {R}} + {}$ ‘every set of reals is Suslin’ is consistent relative to $\mathsf {ZFC} + {}$ ‘there is a cardinal $\lambda $ that is a limit of Woodin cardinals and of $\mathord {<}\lambda $ -strong cardinals’. The $\Sigma ^2_1$ reflection property of our model is proved using genericity iterations as in Neeman [18] and Steel [22].