Trees of manifolds with boundaries are analogues of spaces which where investigated by W. Jakobsche and which are called trees of closed manifolds. W. Jakobsche described them in terms of inverse limits of certain systems of closed manifolds.
Trees of closed manifolds occur as boundaries of nonpositively curved groups and spaces. H. Fisher showed that they occur as boundaries of some right angled Coxeter groups, J. Swiatkowski showed that they occur as boundaries of some negatively curved pseudomanifolds, and I showed that they occur as boundaries of some systolic groups.
In my talk I want to introduce a definition and to prove some basic properties of trees of manifolds with boundaries and to mention about some conjectures concernig occuring of them as boundaries of some negatively curved groups and spaces.