(Joint work with Steve Tschantz) The first "accessibility" questions for finitely generated groups arose from Stallings' splitting theorem for infinite ended groups. Later Dunwoody proved that finitely presented groups are indeed accessible with respect to splittings over finite groups. Bestvina and Feighn followed Dunwoody’s result with an accessibility result for finitely presented groups when splittings subgroups are "small." The notion of JSJ-decompositions of finitely presented groups arose from geometrical/algebraic decompositions of closed 3-manifolds over certain embedded separating surfaces. For 1-ended finitely presented groups Rips and Sela showed the existence of (unique) JSJ decompositions over 2- ended splittings. Dunwoody and Sageev introduced the notion of minimal virtually abelian splitting subgroups into the JSJ arena. We introduce a notion of "strong accessibility" over minimal splittings of groups that naturally generalizes the original accessibility results over finite and small splitting subgroups. We discuss two results. The first is a strong accessibility result for finitely generated Coxeter groups over minimal splittings. The second is a "best possible" JSJ-result for splitting finitely generated Coxeter groups over virtually abelian splitting subgroups. Splittings over minimal virtually abelian subgroups plays an important role in the JSJ result. Both results are "visual" in the sense that the critical decompositions involved can be "seen" geometrically in the presentation diagram of a Coxeter group.