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Let $G$ be a digraph and $A(G)$ be the adjacency matrix of $G$. Let $D(G)$ bethe diagonal matrix with outdegrees of vertices of $G$. For any real$\alpha\in[0,1]$, Liu et al. \cite{LWCL} defined the matrix $A_\alpha(G)$ as$$A_\alpha(G)=\alpha D(G)+(1-\alpha)A(G).$$ The largest modulus of theeigenvalues of $A_\alpha(G)$ is called the $A_\alpha$ spectral radius of $G$.In this paper, we determine the digraphs which attain the maximum (or minimum)$A_\alpha$ spectral radius among all strongly connected digraphs with givenparameters such as girth, clique number, vertex connectivity or arcconnectivity. We also discuss a number of open problems.

Merging the A- and Q-spectral theories for digraphs