My surprise about the frequent mentions of Tarkovsky is not a reflection on the standing i think he should have,(as the fact i list him as one of my favourites makes clear),just that i hadn't realised quite so many people share my opinion.He's certainly a lot less well known in Britain than,say,Bergman,Fellini and Kurosawa,but then the wider public here aren't connoisseurs of World Cinema!Bergman of course considered him "the greatest".Andrei Rublev and Mirror are absolute masterpieces.
I neglected to mention some important American directors:Ford,Hawks,Altman,Scorsese,Allen, and several fine current international directors;Hou Hsiao-Hsien,Yang,Zhang Yimou(though his recent efforts have been relatively mediocre),Almodovar(i'm not his greatest fan,i must admit,but i have enjoyed most of his films)...
Not many mentions of female directors so far, i notice.