  We prove the boundedness theorem for Fano threefolds with log-terminal
singularities of any fixed index. This is an improvement of our earlier result,
where we required additionally that the variety is Q-factorial, with Picard
number 1. The new ideas of the paper include the following.
  1. Using Alexeev Minimal Model program with suitable boundary to find
horizontal extremal contractions.
  2. Using Koll\'ar's effective Base Point Freeness theorem.
  3. Using Kawamata's result on the length of extremal curves with suitable
boundary to avoid gluing curves in some cases.
