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The Hilbert scheme of points of a projective space is central for the study of finite schemes. Its open subset parameterizing Gorenstein subschemes is called the Gorenstein Hilbert scheme of points. In my talk I will present the basics of the theory of finite Gorenstein schemes and explain how it is used in the study of the aforementioned open subset. In particular I will sketch some results on irreducibility, obtained together with G. Casnati and R. Notari.