For moduli problems it is often not so hard to show that they define algebraic stacks, but these are often not separated. This is can sometimes be resolved by showing that some open "semistable" part of the moduli problem admits proper coarse moduli spaces. While we have necessary and sufficient conditions that an open the "semistable" part needs to satisfy to have a proper quotient, much fewer techniques are known to find such open subsets. In the talk I will try to explain how looking at this problem for algebraic stacks, gives an approach to the problem to characterize the open subsets of smooth projective varieties with an action of a torus that admit a proper quotient space.