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Let F/F_0 be an unramified quadratic extension of p-adic fields (p odd). I will introduce related unitary Rapoport—Zink spaces with (maximal) parahoric levels and study their rich geometry, including the Drinfeld half plane as a semi-stable example. I will define several families of special cycles and stratifications on these moduli spaces, which behave well under pullbacks and projections. I will present some local modularity conjectures on "Fourier transforms" of special divisors. Via stratification, they are related to explicit Tate conjectures for curves on explicit Deligne—Lusztig varieties. Finally, I will discuss some arithmetic applications.