  Zariski decompositions play an important role in the theory of algebraic
surfaces. For making geometric use of the decomposition of a given divisor, one
needs to pass to a multiple of the divisor in order to clear denominators. It
is therefore an intriguing question whether the surface has a 'universal
denominator' that can be used to simultaneously clear denominators in all
Zariski decompositions on the surface. We prove in this paper that, somewhat
surprisingly, this condition of bounded Zariski denominators is equivalent to
the bounded negativity of curves that is addressed in the Bounded Negativity
Conjecture. Furthermore, we provide explicit bounds for Zariski denominators
and negativity of curves in terms of each other.
