Publisher Summary This chapter discusses discrete non-euclidean geometry with emphasis on inner product spaces, spherical geometry, elliptic geometry, and hyperbolic geometry. It describes discrete figures of finite dimensional elliptic, spherical, hyperbolic and Euclidean type, starting with the positive definite space Rd and the indefinite space R1,d as a common framework. Finite Euclidean and non-Euclidean sets have been characterized in terms of matrices of inner products or distances. Such matrices and their generalizations also serve the description of certain notions of a combinatorial nature, such as distance spaces, codes, designs, graphs, matroids, root systems, integral lattices, and finite groups. It presents a brief introduction to spherical space Sp-1, elliptic space Ip-1, hyperbolic space Hq, and Euclidean space Eq-1. Rq,p is the R-vector space of dimension p + q provided with a real bilinear form of signature p,q. Vectors x Є Rp,q are called positive, isotropic, negative, according to whether their norm (x,x) is positive, zero, negative, respectively.