In recent years, a novel attitude to the classical problem of identifying and classifying special linear systems in projective n space, has been emerged. For a subvariety Z of the projective n space with defining ideal I, let P_1,�,P_s be general distinct points in this space and let m_1,�,m_s be positive integers which at least one of them is greater than one. On the subspace of those elements of degree d part of the homogeneous ideal I which vanish at P_i with multiplicity at least m_i, each fat point m_iP_i defines a specific number of linear relations on this subspace. For a given set of points P_i with multiplicity m_i, it is expected that these linear equations to be linearly independent. If it is not the case, then one says that the variety Z admits an unexpected hypersurface with respect to the fat point subscheme defined by these fat points, and this linear subspace is called a special linear system on the variety Z. Each element of this subspace defines a hypersurface, known as an unexpected hypersurface. In this talk, we review some interesting examples which brought into the scene with this new approach and explain some existing methods to construct unexpected hypersurfaces.

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