Mori dream spaces form a class of algebraic varieties that play a significant role in birational geometry, as they exhibit ideal behaviour within the minimal model program. In this talk, we discuss the birational geometry of hypersurfaces in products of weighted projective spaces, focusing particularly on cases where they are Mori dream spaces. We generalize the results obtained by J.C. Ottem and, if time permits, we will address the Kawamata-Morrison cone conjecture for certain anticanonical Calabi-Yau hypersurfaces in products of weighted projective spaces.