This is a follow-up study to the Phase 3 pivotal study (281102 \[NCT03393975\]) and will be comprised of two treatment cohorts (Prophylactic and On-demand) consisting of naïve and non-naïve participants who were born with TTP, with a total duration of approximately 6 years. A maximum of approximately 77 participants will be enrolled in this study (approximately 57 participants who have completed study 281102 \[NCT03393975\] \[non-naïve participants\] and at least 20 naïve participants (participants who are naïve to TAK-755). Participants from Expanded Access Programs, participants from study 281102 (NCT03393975) who had an allergic reaction to standard of care treatment, and participants who completed the Phase 1 study (281101 \[NCT02216084\]) but did not participate in Study 281102, will also be eligible for enrollment in this continuation study as naïve participants. TAK-755 is given slowly through a vein (intravenous infusion). Participants in the prophylactic cohort will be able to opt for treatment in a home setting by caregiver or self-infusion subject to conditions and to local regulatory approval.