  We present a canonical way to decompose finite graphs into highly connected
local parts. The decomposition depends only on an integer parameter whose
choice sets the intended degree of locality. The global structure of the graph,
as determined by the relative position of these parts, is described by a
coarser $\it model$. This is a simpler graph determined entirely by the
decomposition, not imposed.
  The model and decomposition are obtained as projections of the tangle-tree
structure of a covering of the given graph that reflects its local structure
while unfolding its global structure. In this way, the tangle theory from graph
minors is brought to bear canonically on arbitrary graphs, which need not be
tree-like.
  Our theorem extends to locally finite quasi-transitive graphs, and in
particular to locally finite Cayley graphs. It thereby offers a canonical
decomposition for finitely generated groups into local parts, whose relative
structure is displayed by a graph.
