  Let $G$ be a digraph with adjacency matrix $A(G)$. Let $D(G)$ be the diagonal
matrix with outdegrees of vertices of $G$. Nikiforov \cite{Niki} proposed to
study the convex combinations of the adjacency matrix and diagonal matrix of
the degrees of undirected graphs. Liu et al. \cite{LWCL} extended the
definition to digraphs. For any real $\alpha\in[0,1]$, the matrix $A_\alpha(G)$
of a digraph $G$ is defined as $$A_\alpha(G)=\alpha D(G)+(1-\alpha)A(G).$$ The
largest modulus of the eigenvalues of $A_\alpha(G)$ is called the $A_\alpha$
spectral radius of $G$, denoted by $\lambda_\alpha(G)$. This paper proves some
extremal results about the spectral radius $\lambda_\alpha(G)$ that generalize
previous results about $\lambda_0(G)$ and $\lambda_{\frac{1}{2}}(G)$.
  In particular, we characterize the extremal digraph with the maximum (or
minimum) $A_\alpha$ spectral radius among all $\widetilde{\infty}$-digraphs and
$\widetilde{\theta}$-digraphs on $n$ vertices. Furthermore, we determine the
digraphs with the second and the third minimum $A_\alpha$ spectral radius among
all strongly connected bicyclic digraphs. For $0\leq\alpha\leq\frac{1}{2}$, we
also determine the digraphs with the second, the third and the fourth minimum
$A_\alpha$ spectral radius among all strongly connected digraphs on $n$
vertices. Finally, we characterize the digraph with the minimum $A_\alpha$
spectral radius among all strongly connected bipartite digraphs which contain a
complete bipartite subdigraph.
