In this talk we discuss the geometric and non-geometric faces of closed string vacua. The associated closed string geometries are described by new non-commutative as well as non-associative algebras, which can be characterized by certain 3-cocycles in Lie algebra cohomology. We present an associated star-product algebra on functions in phase space. We also discuss some aspects of uncertainty relations, as well as the question, if the non-associative structures are visible in conformal field theory and in double field theory.