  We show stability of the $L^2$-projection onto Lagrange finite element spaces
with respect to (weighted) $L^p$ and $W^{1,p}$-norms for any polynomial degree
and for any space dimension under suitable conditions on the mesh grading. This
includes $W^{1,2}$-stability in two space dimensions for any polynomial degree
and meshes generated by newest vertex bisection. Under realistic but
conjectured assumptions on the mesh grading in three dimensions we show
$W^{1,2}$-stability for all polynomial degrees. We also propose a modified
bisection strategy that leads to better $W^{1,p}$-stability. Moreover, we
investigate the stability of the $L^2$-projection onto Crouzeix-Raviart
elements.
