  We briefly review the current status of the algebraic approach to quantum
field theory on globally hyperbolic spacetimes, both axiomatic -- for general
field theories, and constructive -- for a linear Klein-Gordon model. We recall
the concept of F-locality, introduced in the latter context in BS Kay: Rev.
Math. Phys., Special Issue, 167-195 (1992) and explain how it can be formulated
at an axiomatic level for a general field theory (as a condition on
algebras-with-net-structure) on both globally hyperbolic and non globally
hyperbolic spacetimes. We also discuss the current status of the question
whether/when algebras satisfying F-locality can exist for the Klein-Gordon
model on spacetimes which are chronology violating.
