  In this article the $(b, c)$-inverse will be studied. Several equivalent
conditions for the existence of the $(b,c)$-inverse in rings will be given. In
particular, the conditions ensuring the existence of the $(b,c)$-inverse, of
the annihilator $(b,c)$-inverse and of the hybrid $(b,c)$-inverse will be
proved to be equivalent, provided $b$ and $c$ are regular elements in a unitary
ring $R$. In addition, the set of all $(b,c)$-invertible elements will be
characterized and the reverse order law will be also studied. Moreover, the
relationship between the $(b,c)$-inverse and the Bott-Duffin inverse will be
considered. In the context of Banach algebras, integral, series and limit
representations will be given. Finally the continuity of the $(b,c)$-inverse
will be characterized
