For a string w over an alphabet Σ, we consider a composite data structure called the all-suffixes directed acyclic word graph (ASDAWG). ASDAWG(w) has |w| + 1 initial nodes, and the dag induced by all reachable nodes from the k-th initial node conforms with DAWG(w[k:]), where w[k:] denotes the k-th suffix of w. We prove that the size of the minimum ASDAWG(w) (MASDAWG(w)) is Θ(|w|) for |Σ| = 1, and is Θ(|w|2) for |Σ|≥ 2. Moreover, we introduce an on-line algorithm which directly constructs MASDAWG(w) for given w, whose running time is linear with respect to its size. We also demonstrate some application problems, beginning-sensitive pattern matching, region-sensitive pattern matching, and VLDC-pattern matching, for which AS-DAWGs are useful.