In this paper, we study a kind of dual generalized inverse, which is called the dual Drazin inverse. Unlike the real matrices case, the dual Drazin inverse of a square dual matrix may not exist. It is shown that the dual Drazin inverse is unique when it exists. Some necessary and sufficient conditions for the existence of the dual Drazin inverse are presented. A compact formula for the computation of the dual Drazin inverse is given when it exists. Moreover, we find an unexpected result that the dual Drazin inverse can be obtained by computing the Drazin inverse of a 2 × 2 upper triangular block matrix. We also introduce the dual Drazin-inverse solution of systems of linear dual equations. Some characterizations of the dual Drazin-inverse solution are given. In addition, some numerical examples are provided to illustrate the results.