We introduce and study problems of distributed observation with bounded or unbounded memory. We are given a system modeled as a finite-word language L over some fi- nite alphabet ? and subalphabets \sum _{1, \ldots ,} \sum _n of ? modeling n distinct observation points. We want to build (when there exist) n observers which collect projections of a behavior in L onto \sum _{1, \ldots ,} \sum _n then send them to a central decision point. The latter must determine whether the original behavior was in a given K ? L. In the unbounded-memory case, observers record the entire sequence they observe. In the bounded-memory case, they are required to be finitestate automata. We show that, when L is trace-closed with respect to the usual dependence relation induced by \sum _{1, \ldots ,} \sum _n unbounded-memory observability is equivalent to K being centrally observable and trace-closed, thus decidable. When L is not trace-closed, the problem is undecidable, even if K and L are regular. We also show that boundedmemory observability is equivalent to unbounded-memory observability (thus decidable) when L is trace-closed and \sum {_i } are pairwise disjoint. Otherwise, the problem remains open. In the decidable cases, observers and decision function can be automatically synthesized.