  We show that the moduli stacks of Bridgeland semistable objects on smooth
projective 3-folds are proper algebraic stacks of finite type, if they satisfy
the Bogomolov-Gieseker (BG for short) inequality conjecture proposed by Bayer,
Macr\`i and the second author. The key ingredients are the equivalent form of
the BG inequality conjecture and its generalization to arbitrary very weak
stability conditions. This result is applied to define Donaldson-Thomas
invariants counting Bridgeland semistable objects on smooth projective
Calabi-Yau 3-folds satisfying the BG inequality conjecture, for example on
\'etale quotients of abelian 3-folds.
