Abstract: Every complex quasi-projective surface Y has a completion (X,D), where X is projective, X\D=Y and D is a reduced snc-divisor, the curve
'at infinity'. Problems concerning Y can be successfully approached by combining the logarithmic minimal model program applied to (X,D) with known structure theorems and inequalities for relatively minimal models. While the log MMP works equally well for log surfaces (X,D) where D is a Q-divisor, it does not have such a nice geometric interpretation. Still,
we will show that one can use it to solve some open problems (for example to understand the geometry of cuspidal curves in P^2).