  In this article we review our recent work on the causal structure of
symmetric spaces and related geometric aspects of Algebraic Quantum Field
Theory. Motivated by some general results on modular groups related to nets of
von Neumann algebras,we focus on Euler elements of the Lie algebra, i.e.,
elements whose adjoint action defines a 3-grading. We study the wedge regions
they determine in corresponding causal symmetric spaces and describe some
methods to construct nets of von Neumann algebras on causal symmetric spaces
that satisfy abstract versions of the Reeh--Schlieder and the
Bisognano-Wichmann condition.
