A Phase 3b, open-label, single-arm, rollover study to evaluate the long-term safety of luspatercept, to the following participants: - Participants receiving luspatercept on a parent protocol at the time of their transition to the rollover study, who tolerate the protocol-prescribed regimen in the parent trial and, in the opinion of the investigator, may derive clinical benefit from continuing treatment with luspatercept - Participants in the follow-up phase previously treated with luspatercept or placebo in the parent protocol will continue into long-term post-treatment follow-up in the rollover study until the follow-up commitments are met - The study design is divided into the Transition Phase, Treatment Phase and Follow-up Phase. Participants will enter transition phase and depending on their background will enter either the treatment phase or the Long-term Post-treatment Follow-up (LTPTFU) phase - Transition Phase is defined as one Enrollment visit - Treatment Phase: For participants in luspatercept treatment the dose and schedule of luspatercept in this study will be the same as the last dose and schedule in the parent luspatercept study. This does not apply to participants that are in long-term follow-up from the parent protocol - Follow-up Phase includes: - 42 Day Safety Follow-up Visit - During the Safety Follow up, the participants will be followed for 42 days after the last dose of luspatercept, for the assessment of safety-related parameters and adverse event (AE) reporting - Long-term Post-treatment Follow-up (LTPTFU) Phase - Participants will be followed for overall survival every 6 months for at least 5 years from first dose of luspatercept in the parent protocol, or 3 years of post-treatment from last dose, whichever occurs later, or until death, withdrawal of consent, study termination, or until a subject is lost to follow-up. Participants will also be monitored for progression to AML or any malignancies/pre-malignancies. New anticancer or disease related therapies should be collected at the same time schedule Participants transitioning from a parent luspatercept study in post-treatment follow-up (safety or LTPTFU) will continue from the same equivalent point in this rollover study. The rollover study will be terminated, and relevant participants will discontinue from the study when all participants fulfill at least 5 years from the first dose of luspatercept in the parent protocol, or 3 years of post-treatment from last dose, whichever occurs later.