  We investigate the horizon structure of the rotating Einstein-Born-Infeld
solution which goes over to the Einstein-Maxwell's Kerr-Newman solution as the
Born-Infeld parameter goes to infinity ($\beta \rightarrow \infty$). We find
that for a given $\beta$, mass $M$ and charge $Q$, there exist critical
spinning parameter $a_{E}$ and $r_{H}^{E}$, which corresponds to an extremal
Einstein-Born-Infeld black hole with degenerate horizons, and $a_{E}$ decreases
and $r_{H}^{E}$ increases with increase in the Born-Infeld parameter $\beta$.
While $a<a_{E}$ describe a non-extremal Einstein-Born-Infeld black hole with
outer and inner horizons. Similarly, the effect of $\beta$ on infinite redshift
surface and in turn on ergoregion is also included. It is well known that a
black hole can cast a shadow as an optical appearance due to its strong
gravitational field. We also investigate the shadow cast by the non-rotating
($a=0$) Einstein-Born-Infeld black hole and demonstrate that the null geodesic
equations can be integrated that allows us to investigate the shadow cast by a
black hole which is found to be a dark zone covered by a circle. Interestingly,
the shadow of the Einstein-Born-Infeld black hole is slightly smaller than for
the Reissner-Nordstrom black hole. F urther, the shadow is concentric circles
whose radius decreases with increase in value of parameter $\beta$.

General Relativity and Quantum Cosmology;High Energy Astrophysical Phenomena