  The Welschinger invariants of real rational algebraic surfaces count real
rational curves which represent a given divisor class and pass through a
generic conjugation-invariant configuration of points. No invariants counting
real curves of positive genera are known in general. We indicate particular
situations, when Welschinger-type invariants counting real curves of positive
genera can be defined. We also prove the positivity and give asymptotic
estimates for such Weischinger-type invariants for several del Pezzo surfaces
of degree $\ge2$ and suitable real nef and big divisor classes. In particular,
this yields the existence of real curves of given genus and of given divisor
class passing through any appropriate configuration of real points on the given
surface.
