  In this paper we study very smooth points of Banach spaces with special
emphasis on spaces of operators. We show that when the space of compact
operators is an $M$-ideal in the space of bounded operators, a very smooth
operator $T$ attains its norm at a unique vector $x$ (up to a constant
multiple) and $T(x)$ is a very smooth point of the range space. We show that if
for every equivalent norm on a Banach space, the dual unit ball has a very
smooth point then the space has the Radon--Nikod\'{y}m property. We give an
example of a smooth Banach space without any very smooth points.
