1.The osserman conditions in Semi-Riemannian geometry. 2.The osserman conjecture in Riemannian geometry. 3.Lorentzian osserman manifolds. 4.Four-Dimensional semi-Riemannian osserman manifolds with metric tensors of signature (2,2). 5.Semi-Riemannian osserman manifolds. 6.Generalizations and osserman-Related conditions.