A log Del Pezzo surface is a normal log terminal surface with anti-ample canonical bundle. Over the complex numbers, Keel and McKernan have classified all but a bounded family of the simply connected log Del Pezzo surfaces of rank one. In this talk we extend their classification in positive characteristic, and in particular we prove that for $p>5$ every log Del Pezzo surface of rank one lifts to characteristic zero with smooth base. As a consequence, we see that Kawamata-Viehweg vanishing holds in this setting. Finally, we exhibit some counter-examples in characteristic two, three and five.