Gutman and Wagner proposed the concept of matching energy (ME) and pointed out that the chemical applications of ME go back to the 1970s. Let $G$ be a simple graph of order $n$ and $mu_1,mu_2,ldots,mu_n$ be the roots of its matching polynomial. The m
atching energy of $G$ is defined to be the sum of the absolute values of $mu_{i} (i=1,2,ldots,n)$. In this paper, we characterize the graphs with minimal matching energy among all unicyclic and bicyclic graphs with a given diameter $d$.
