  We initiate a systematic quantitative study of subsets of rational points
that are integral with respect to a weighted boundary divisor on Fano
orbifolds. We call the points in these sets Campana points. Earlier work of
Campana and subsequently Abramovich shows that there are several reasonable
competing definitions for Campana points. We use a version that delineates well
different types of behaviour of points as the weights on the boundary divisor
vary. This prompts a Manin-type conjecture on Fano orbifolds for sets of
Campana points that satisfy a klt (Kawamata log terminal) condition. By
importing work of Chambert-Loir and Tschinkel to our set-up, we prove a log
version of Manin's conjecture for klt Campana points on equivariant
compactifications of vector groups.
