Inclusion Criteria:
Participants who have completed TAK-755 Phase 3 pivotal Study 281102 (NCT03393975) in the prophylactic cohort and who meet all of the following criteria are eligible for this study:
* Participants or legally authorized representative has provided signed informed consent \>=18 years of age and/or assent form \<18 years of age.
* Participant 0 to 70 years of age at the time of screening of the 281102 (NCT03393975) study.
* Participant has been diagnosed with severe congenital ADAMTS-13 deficiency.
* Participant does not display any severe thrombotic thrombocytopenic purpura (TTP) signs (platelet count \<100,000/ microliter (mcL) and elevation of lactate dehydrogenase (LDH) greater than (\>) 2 × ULN at screening (prophylactic cohort only).
* Participants \>=16 years of age must have a Karnofsky score \>= 70% and participants \<16 years of age must have a Lansky score \>=80%.
* If female of childbearing potential, participant presents with a negative serum or urine pregnancy test confirmed not more than 7 days before the first IP administration and agrees to employ adequate birth control measures for the duration of the study and to undergo quarterly pregnancy testing.
* Sexually active males must use an accepted and effective method of contraception during the treatment and until a minimum of 16 days after the last dose administered.
* Participant is willing and able to comply with the requirements of the protocol.
All naïve participants and non-naïve on-demand cohort participants:
Naïve participants can only be enrolled in this continuation after enrollment of the adult participants in the prophylactic arm of TAK-755 Phase 3 pivotal study 281102 (NCT03393975) has been completed. Naïve pediatric participants can be enrolled after enrollment of the respective age cohort into the pivotal Phase 3 study 281102 (NCT03393975) has been completed. The following criteria also applies to participants who completed study 281101 (NCT02216084), but did not participate in 281102 (NCT03393975).
Naïve participants and participants who were enrolled into the on-demand cohort of theTAK-755 Phase 3 pivotal study 281102 (NCT03393975) who meet ALL of the following criteria are eligible for this study:
* Participant is naïve or was enrolled into the on-demand cohort of the TAK-755 Phase 3 pivotal study 281102 (NCT03393975) for treatment of an acute TTP event but did not receive prophylactic treatment.
* Participant or legally authorized representative has provided signed informed consent (\>=18 years of age) and/or assent form (\<18 years of age).
* Participant is 0 to 70 years of age at the time of screening.
* Participant has been diagnosed with severe congenital ADAMTS-13 deficiency defined as:
* Confirmed by molecular genetic testing, documented in participant history or at screening, and
* ADAMTS-13 activity \<10% as measured by the fluorescence resonance energy transfer (FRETS)-VWF73 assay, documented in participant history or at screening. Participants currently receiving standard of care prophylactic therapy may exceed 10% ADAMTS-13 activity at screening.
* Participants currently receiving prophylactic therapy will be screened immediately prior to their usual prophylactic infusion.
* Participant does not display any severe TTP signs (platelet count \<100,000/microliter (mcL) and elevation of LDH \>2 × ULN) at screening (prophylactic cohort only).
* Participants \>=16 years of age must have a Karnofsky score \>=70% and participants \<16 years of age must have a Lansky score \>=80%.
* Participants is hepatitis C virus negative (HCV-) as confirmed by antibody or polymerase chain reaction testing OR HCV positive (HCV+) if their disease is chronic but stable.
* If female of childbearing potential, participant presents with a negative serum or urine pregnancy test confirmed not more than 7 days before the first IP administration and agrees to employ adequate birth control measures for the duration of the study and to undergo quarterly pregnancy testing.
* Sexually active males must use an accepted and effective method of contraception during treatment and until a minimum of 16 days after the last dose administered.
* Participant is willing and able to comply with the requirements of the protocol.
Participants from an Expanded Access Program or participants in Study 281102 (NCT03393975) who had an allergic reaction to standard of care prophylactic treatment must meet all of the following criteria:
Participants from an expanded access program as well as participants who participated in Study 281102(NCT03393975) who had an allergic reaction to standard-of-care prophylactic treatment are eligible for enrollment in the continuation study if they meet ALL of the following criteria.
* Participants or legally authorized representative has provided signed informed consent (\>=18 years of age) and/or assent (\<18 years of age).
* Participants is 0 to 70 years of age at the time of screening.
* Participants has been diagnosed with severe congenital ADAMTS-13 deficiency defined as:
* Confirmed by molecular genetic testing, documented in participant history or at screening, and
* ADAMTS-13 activity \<10% as measured by the fluorescence resonance energy transfer (FRETS)- VWF 73 assay, documented in participant history or at screening. Participants currently receiving standard of care prophylactic therapy may exceed 10% ADAMTS 13 activity at screening.
* Participant does not display any severe TTP signs (platelet count \<100,000/mcL and elevation of LDH \>2 × ULN) at screening (prophylactic cohort only).
* Participants \>=16 years of age must have a Karnofsky score \>=70% and participants \<16 years of age must have a Lansky score \>=80%.
* If female of childbearing potential, participant presents with a negative serum or urine pregnancy test confirmed not more than 7 days before the first IP administration and agrees to employ highly effective birth control measures for the duration of the study and to undergo quarterly pregnancy testing.
* Sexually active males must use an accepted and effective method of contraception during treatment and until a minimum of 16 days after the last dose administered.
* Participant is willing and able to comply with the requirements of the protocol.
Exclusion Criteria:
Participants who have completed TAK-755 Phase 3 pivotal study (281102) (NCT03393975) and naïve participants and non-naïve on-demand cohort participants and participants from an Expanded Access Program or participants in Study 281102 (NCT03393975) who had an allergic reaction to standard-of-care prophylactic treatment. The following criteria also applies to participants who completed study 281101 (NCT02216084), but did not participate in 281102 (NCT03393975).
* Participant has been diagnosed with any other TTP-like disorder (microangiopathic hemolytic anemia), including immune-mediated TTP.
* Known life-threatening hypersensitivity reaction, including anaphylaxis, to the parent molecule ADAMTS-13, hamster protein, or other constituents of TAK-755.
* Participant has a presence of a functional ADAMTS-13 inhibitor at screening.
* Participant has a medical history of a genetic or acquired immune deficiency that would interfere with the assessment of product immunogenicity, including participants who are human immunodeficiency virus-positive with an absolute cluster of differentiation 4 (CD4) count \< 200/ cubic millimeter (mm\^3) or who are receiving chronic immunosuppressive drugs.
* Participant has a history of significant neurological events, such as major stroke, indicating that a relapse might have severe consequences, as judged by the investigator.
* Participant has been diagnosed with severe cardiovascular disease (New York Heart Association classes 3 to 4).
* Participant with end stage renal disease requiring chronic dialysis.
* Participant has been diagnosed with hepatic dysfunction, as evidenced by, but not limited to, any of the following:
* Serum alanine aminotransferase \>= 2 × ULN
* Severe hypoalbuminemia \<24 gram per liter (g/L)
* Portal vein hypertension (e.g., presence of otherwise unexplained splenomegaly, history of esophageal varices).
* In the opinion of the investigator, the participant has another clinically significant concomitant disease that may pose additional risks for the participant.
* Participant has been treated with an immunomodulatory drug, excluding topical treatment (e.g., ointments, nasal sprays), within 30 days prior to enrollment. Use of corticosteroids in conjunction with administration of fresh frozen plasma to prevent allergic reactions is permitted.
* Participant has an acute illness (e.g., influenza, flu-like syndrome, allergic rhinitis/conjunctivitis, bronchial asthma) at the time of screening (prophylactic cohort only).
* Participant is receiving or anticipates receiving another investigational drug and/or interventional drug within 30 days before enrollment.
* Participant has a history of drug and/or alcohol abuse within the last 2 years.
* Participant has a progressive fatal disease and/or life expectancy of \<= 3 months.
* Participant is identified by the investigator as being unable or unwilling to cooperate with study procedures.
* Participant suffers from a mental condition rendering him/her unable to understand the nature, scope, and possible consequences of the study and/or evidence of an uncooperative attitude.
* Participant is a family member or employee of the sponsor or investigator.
* If female, participant is pregnant or lactating at the time of enrollment.
* In the UK only: Participants who have not previously received a dose of TAK-755.
