  We prove that the maximum number $N_c$ of non-relativistic electrons that a
nucleus of charge $Z$ can bind is less than $1.22Z+3Z^{1/3}$. This improves
Lieb's upper bound $N_c<2Z+1$ [{\it Phys. Rev. A} {\bf 29}, 3018-3028 (1984)]
when $Z\ge 6$. Our method also applies to non-relativistic atoms in magnetic
field and to pseudo-relativistic atoms. We show that in these cases, under
appropriate conditions, $\limsup_{Z\to \infty}N_c/Z \le 1.22$.
