Summary: The square \(G^{2}\) of a graph \(G\) is the graph having the same vertex set as \(G\) and two vertices are adjacent if and only if they are at distance at most 2 from each other. It is known that if \(G\) has no cut-vertex, then \(G^{2}\) is Hamilton-connected (see [G. Chartrand, A. M. Hobbs, H. A. Jung, S. F. Kapoor, C. St. J. A. Nash-Williams, “The square of a block is hamiltonian connected,” J. Comb. Theory, Ser. B 16, 290–292 (1974; Zbl 0277.05129); R.J. Faudree and R.H. Schelp, The square of a block is strongly path connected, J. Comb. Theory, Ser. B 20, 47–61 (1976; Zbl 0284.05116)]). We prove that if \(G\) has only one cut-vertex, then \(G^{2}\) is Hamilton-connected. In the case that \(G\) has only two cut-vertices, we prove that if the block that contains the two cut-vertices is hamiltonian, then \(G^{2}\) is Hamilton-connected. Further, we characterize all graphs with at most one cycle having Hamilton-connected square.