either a compact simple Lie group, a Grassmannian, a Lagrangian Grassmannian, or a double Lagrangian Grassmannian of subspaces of for normed division algebras A and B. A similar construction produces the irreducible non-compact Riemannian symmetric spaces. General symmetric spaces An important class of symmetric spaces generalizing the Riemannian symmetric spaces are pseudo-Riemannian symmetric spaces, in which the Riemannian metric is replaced by a pseudo-Riemannian metric (nondegenerate instead of positive definite on each tangent space). In particular, Lorentzian symmetric spaces, i.e., n dimensional pseudo-Riemannian symmetric spaces of signature (n − 1,1), are important in general relativity, the most notable examples being Minkowski space, De Sitter space and anti-de Sitter space (with zero, positive and negative curvature respectively). De Sitter space of dimension n may be identified with the 1-sheeted hyperboloid in a Minkowski space of dimension n + 1. Symmetric and locally symmetric spaces in general can be regarded as affine symmetric spaces. If M = G/H is a symmetric space, then Nomizu showed that there is a G-invariant torsion-free affine connection (i.e. an affine connection whose torsion tensor vanishes) on M whose curvature is parallel. Conversely a manifold with such a connection is locally symmetric (i.e., its universal cover is a symmetric space). Such manifolds can also be described as those affine manifolds whose geodesic symmetries are all globally defined affine diffeomorphisms, generalizing the Riemannian and pseudo-Riemannian case. Classification results The classification of Riemannian symmetric spaces does not extend readily to the general case for the simple reason that there is no general splitting of a symmetric space into a product of irreducibles. Here a symmetric space G/H with Lie algebra is said to be irreducible if is an irreducible representation of . Since is not semisimple (or even reductive) in general, it can have indecomposable representations which are not irreducible. However, the irreducible symmetric spaces can be classified. As shown by Katsumi Nomizu, there is a dichotomy: an irreducible symmetric space G/H is either flat (i.e., an affine space) or is semisimple. This is the analogue of the Riemannian dichotomy between Euclidean spaces and those of compact or noncompact type, and it motivated M. Berger to classify semisimple symmetric spaces (i.e., those with semisimple) and determine which of these are irreducible. The latter question is more subtle than in the Riemannian case: even if is simple, G/H might not be irreducible. As in the Riemannian case there are semisimple