  Among geometrically rational surfaces, del Pezzo surfaces of degree two over
a field k containing at least one point are arguably the simplest that are not
known to be unirational over k. Looking for k-rational curves on these
surfaces, we extend some earlier work of Manin on this subject. We then focus
on the case where k is a finite field, where we show that all except possibly
three explicit del Pezzo surfaces of degree two are unirational over k.
