  We present new upper and lower bounds for the numerical radius of a bounded
linear operator defined on a complex Hilbert space, which improve on the
existing bounds. Among many other inequalities proved in this article, we show
that for a non-zero bounded linear operator $T$ on a Hilbert space $H,$
$w(T)\geq \frac{\|T\|}{2}+\frac{m(T^2)}{2\|T\|}, $ where $w(T)$ is the
numerical radius of $T$ and $m(T^2)$ is the Crawford number of $T^2$. This
substantially improves on the existing inequality $w(T)\geq \frac{\|T\|}{2} .$
We also obtain some upper and lower bounds for the numerical radius of operator
matrices and illustrate with numerical examples that these bounds are better
than the existing bounds.
