A one-way infinite chain in a graph is a sequence \(X_0e_0X_1e_1,\ldots\) where \(X_0,X_1,\ldots\) are vertices of \(G\), \(e_0,e_1,\ldots\) are distinct edges of \(G\) and \(e_i\) joins \(X_i\) and \(X_{i+1}\) for \(i=0,1,\ldots\). A two-way infinite chain and a finite chain are defined analogously. The graphs that can be decomposed into two-way infinite chains and into finite closed chains were characterized by \textit{C. St. J. A. Nash-Williams} [Proc. Lond. Math. Soc. 10, 221--238 (1960; Zbl 0095.37901)]. The author proves that any connected graph has a decomposition into chains such that at most one of these is one-way infinite and each vertex is the end vertex of at most one of the chains and no vertex of infinite degree is such an end vertex. He also characterizes graphs that have a decomposition into one-way infinite chains. Both of these problems were raised by \textit{R. B. Eggleton} and \textit{D. K. Skilton} [Graph theory, Proc. 1st Southeast Asian Colloq., Singapore 1983, Lect. Notes Math. 1073, 294--306 (1984; Zbl 0544.05042)].