Summary: An antimagic labeling of a graph with \(M\) edges and \(N\) vertices is a bijection from the set of edges to the set \(\{1,2,3,\dots ,M\}\) such that all the \(N\) vertex-sums are pairwise distinct, where the vertex-sum of a vertex \(v\) is the sum of labels of all edges incident with \(v\). A graph is called antimagic if it has an antimagic labeling. The antimagicness of the Cartesian product of graphs in several special cases has been studied [T.-M. Wang, “Toroidal grids are anti-magic”, Lect. Notes Comput. Sci. 3595, 671–679 (2005; Zbl 1128.05311); Y. Cheng, “A new class of antimagic Cartesian product graphs”, Discrete Math. 308, No. 24, 6441–6448 (2008; Zbl 1211.05140)]. In this paper, we develop new construction methods that are applied to more general cases. We prove that the Cartesian product of paths is antimagic, if one of them has at least three edges. This (almost) answers the open problems in [Y. Cheng, “Lattice grids and prisms are antimagic”, Theor. Comput. Sci. 374, No. 1–3, 66–73 (2007; Zbl 1162.05040)]. We also prove that the Cartesian product of an antimagic regular graph and a connected graph is antimagic, which extends the results of the latter of the two references, where several special cases are studied.