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 complex Hilbert space H, w(T) ≥ ||T2 || + m2||(TT2 || ) , where w(T) is the numerical radius of T and m(T2) is the Crawford number of T2 . This substantially improves on the existing inequality w(T) ≥ ||T2 ||. 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.