We introduce four classes of domino plane partitions and give two weights, i.e., $t$-weight and $\tau$-weight. We observe that many interesting integer sequences related to alternating sign matrices and plane partition appear by specializing the parameters $t$ and $\tau$. We obtain determinantal expressions for the generating functions, but the evaluation of the determinants are not completed. Hence I state several conjectures related to the generating functions of these classes of domino plane partitions.