Hide scroll bar, but while still being able to scroll

I want to be able to scroll through the whole page, but without the scrollbar being shown.

In Google Chrome it’s:

::-webkit-scrollbar {
    display: none;
}

But Mozilla Firefox and Internet Explorer don’t seem to work like that.

I also tried this in CSS:

overflow: hidden;

That does hide the scrollbar, but I can’t scroll any more.

Is there a way I can remove the scrollbar while still being able to scroll the whole page?

With just CSS or HTML, please.