  We obtain upper and lower bounds for the Davis-Wielandt radius of bounded
linear operators defined on a complex Hilbert space, which improve on the
existing ones. We also obtain bounds for the Davis-Wielandt radius of operator
matrices. We determine the exact value of the Davis-Wielandt radius of two
special type of operator matrices $\left(\begin{array}{cc}
  I & B
  0 & 0
  \end{array}\right)$ and $\left(\begin{array}{cc}
  0 & A
  B & 0
  \end{array}\right)$, where $A,B\in \mathcal{B}(\mathcal{H})$, $I$ and $0$ are
the identity operator and the zero operator on $\mathcal{H},$ respectively.
Finally we obtain bounds for the Davis-Wielandt radius of operator matrices of
the form $\left(\begin{array}{cc}
  A& B
  0 & C
  \end{array}\right),$ where $A,B, C\in \mathcal{B}(\mathcal{H}).$
