  Existence criteria for the $(b,c)$-inverse are given.% in terms of
annihilators. We present explicit expressions for the $(b,c)$-inverse by using
inner inverses. We answer the question when the $(b,c)$-inverse of $a\in R$ is
an inner inverse of $a$. As applications, we give a unified theory of some
well-known results of the $\{1,3\}$-inverse, the $\{1,4\}$-inverse, the
Moore-Penrose inverse, the group inverse and the core inverse.
