  Bounds are obtained for the $L^p$ norm of the torsion function $v_{\Omega}$,
i.e. the solution of $-\Delta v=1,\, v\in H_0^1(\Omega),$ in terms of the
Lebesgue measure of $\Omega$ and the principal eigenvalue $\lambda_1(\Omega)$
of the Dirichlet Laplacian acting in $L^2(\Omega)$. We show that these bounds
are sharp for $1\le p\le 2$.
