The purpose of this study is to assess the efficacy and safety of trabectedin+DOXIL as a third-line chemotherapy regimen (treatment) in patients with platinum-sensitive advanced-relapsed epithelial ovarian, primary peritoneal, or fallopian tube cancer who received 2 previous lines of platinum-based chemotherapy.

This is a randomized (individuals assigned to study treatment by chance), open - label (identity of assigned study drug will be known), active - controlled study in adult female patients with platinum-sensitive advanced - relapsed epithelial ovarian, primary peritoneal, or fallopian tube cancer who received 2 previous lines of platinum - based chemotherapy. Approximately 670 participants will be enrolled. Patients will be stratified by 4 criteria defined in the protocol and randomly assigned in a 1:1 ratio to the trabectedin+DOXIL combination therapy group (Arm A) or to the DOXIL (pegylated liposomal doxorubicin) monotherapy group (Arm B). During the treatment phase, patients will receive study drug infusions according to 21 - day cycles in Arm A and 28 - day cycles in Arm B. Treatment will continue until the occurrence of disease progression or unacceptable treatment toxicity, or until 2 cycles after assessment of a complete response (CR). Efficacy assessments will be evaluated using Response Evaluation Criteria in Solid Tumors. Disease assessments, including assessments for patients who discontinue treatment for reasons other than disease progression, will be performed until disease progression, the start of subsequent anticancer therapy, withdrawal of consent, or the clinical cutoff date. Collection of survival status will continue until at least 514 deaths have been observed or until the clinical data cutoff date. Serial pharmacokinetic (PK) samples will be collected in a subset of patients who voluntarily consent to the PK portion of the study. Safety will be monitored throughout the study. An interim analysis of overall survival (OS) will be performed after approximately 308 participants have died. The final analysis of OS will occur when approximately 514 deaths have been observed or until the clinical cutoff date. As of Amendment 6, no new participants will be randomized to study treatment, and treatment with trabectedin should be immediately discontinued for participants assigned to Arm A (trabectedin+DOXIL). All study participants (Arm A or Arm B) currently on study who, in the opinion of the investigator, are deriving clinical benefit may continue treatment with single-agent DOXIL as per the local standard of care.

Inclusion Criteria:

• Histologically proven advanced-relapsed epithelial ovarian, primary peritoneal, or fallopian tube cancer
• Eastern Cooperative Oncology Group (ECOG) performance status grade of 0 or 1
• Received first-line treatment with a platinum-based regimen and had no evidence of disease progression for >= 6 months after the last dose
• Received second-line treatment with a platinum-based regimen, with progression of disease after attaining a response
• Progression of disease based on imaging after the second-line platinum-based regimen (individuals treated with a pegylated liposomal doxorubicin-containing regimen as a second-line therapy are eligible if subsequent disease progression occurs >=9 months from the first dose)
• Evidence of measurable disease at screening as evaluated by Response Evaluation Criteria in Solid Tumors (RECIST) (Version 1.1)
• Participants no longer need to be able to receive intravenous (IV) dexamethasone or an equivalent IV corticosteroid
• Have a known BRCA 1/2 mutation status (for participants who do not have a known BRCA 1/2 status at screening, a blood sample will be collected to determine the status with the results available prior to randomization
• Laboratory values within protocol -defined parameters
• Have left ventricular ejection fraction by multigated acquisition scan (MUGA) scan or 2D-ECHO within normal limits for the institution
• Have side effects (except alopecia) of prior treatment resolved to at least Grade 1 according to the National Cancer Institute - Common Terminology Criteria of Adverse Events (NCICTCAE) (Version 4.0)
• Have a negative urine or serum pregnancy test at screening
• Agrees to protocol-defined use of effective contraception

Exclusion Criteria:

• Diagnosis of ovarian carcinoma with mucinous histology
• Had more than 2 prior lines of systemic therapy. Maintenance therapies and hormonal therapies are not considered additional lines of therapy
• Participants who had a prior exposure to trabectedin or hypersensitivity to any of the excipients will not be excluded from receiving single-agent Doxil
• Prior treatment with doxorubicin or other anthracycline at cumulative doses greater than 300 mg/m2 (calculated using doxorubicin equivalent doses: 1 mg doxorubicin = 1 mg Doxil/Caelyx = 1.8 mg epirubicin = 0.3 mg mitoxantrone = 0.25 mg idarubicin)
• Participants unwilling or unable to have a central venous catheter placed will not be excluded from receiving single-agent Doxil
• Pregnant or breast-feeding
• Would receive study treatment within 3 weeks from radiation therapy, experimental therapy, hormonal therapy, prior chemotherapy, or biological therapy; use an invasive investigational device; or is currently enrolled in an investigational study
• History of another invasive malignancy (except non-metastatic basal cell carcinoma or squamous cell carcinoma of the skin or cervical carcinoma in situ adequately treated) unless in remission for >=5 years, or a non - invasive malignancy requiring ongoing therapy
• Known allergies, hypersensitivity, or intolerance to Doxil, dexamethasone, or their excipients
• Known history of central nervous system metastasis
• Known significant chronic liver disease, such as cirrhosis or active hepatitis (potential participants who test positive for hepatitis B surface antigen or hepatitis C antibodies are allowed provided they do not have active disease requiring antiviral therapy)
• Had a myocardial infarct within 6 months before enrollment, New York Heart Association (NYHA) Class II or greater heart failure, uncontrolled angina, severe uncontrolled ventricular arrhythmias, clinically significant pericardial disease, or electrocardiographic evidence of acute ischemic or active conduction system abnormalities
• Has any of the following medical conditions: uncontrolled diabetes, psychiatric disorder (including dementia) that prevents compliance with protocol, uncontrolled seizures, newly diagnosed deep vein thrombosis, active systemic infection that is likely to interfere with study procedure or results
• Has any condition that, in the opinion of the investigator, would compromise the well-being of the participant or the study or prevent the participant from meeting or performing study requirements