  We compute the purely real Welschinger invariants, both original and
modified, for all real del Pezzo surfaces of degree at least 2. We show that
under some conditions, for such a surface $X$ and a real nef and big divisor
class $D$, through any generic collection of $-DK_X-1$ real points lying on a
connected component of the real part of $X$ one can trace a real rational curve
$C\in|D|$. This is derived from the positivity of appropriate Welschinger
invariants. We furthermore show that these invariants are asymptotically
equivalent, in the logarithmic scale, to the corresponding genus zero
Gromov-Witten invariants. Our approach consists in a conversion of
Shoval-Shustin recursive formulas counting complex curves on the plane blown up
at seven points and of Vakil's extension of the Abramovich-Bertram formula for
Gromov-Witten nvariants into formulas computing real enumerative invariants.
