The Cartesian product of two graphs forms a special class of graphs. First,for a given tree through its Cartesian products withcycles, we discuss its Hamiltonicity and edge-Hamiltonicity. Second, for a given tree through its Cartesian products with paths, we discuss its Hamiltonicity and even-pancyclicity. We find several Hamiltonian graphs in the case that the tree has a perfect matching or a path factor. Some well-known results which have been proved are also given in this thesis with modified results or new approach of proofs. In particular, we prove that the two conditions Hamiltonian and 1-tough are equivalent in those graphs we discussed.