Inclusion criteria:  Female aged ≥ 18 years. Histologically or cytologically documented invasive epithelial ovarian cancer, primary peritoneal carcinoma, or fallopian tube cancer Platinum partially sensitive patients or fully platinum sensitive patients not able to receive or not willing to receive other platinum treatments, who have previously received pegylated liposomal doxorubicin (carboplatin- pegylated liposomal doxorubicin or pegylated liposomal doxorubicin alone). Eastern Cooperative Oncology Group (ECOG) Performance Status (PS) 0- 1. Subject has radiographically-documented measurable disease, as per RECIST 1.1 at study entry (CA-125 rise not supported by radiological evidence of disease is not accepted as criteria for defining progression). Be able to receive IV dexamethasone or an equivalent IV corticosteroid. Have all of the following:  hemoglobin ≥9 g/dL (without transfusion in the prior 7 days). Subjects may be enrolled into the study while receiving recombinant erythropoietin provide they have received recombinant erythropoietin for at least 4 weeks. before the first dose of study drug.  albumin >25 g/L absolute neutrophil count (ANC) >1,500/μL platelet count >100,000/μL (without transfusion in the prior 7 days) either a serum creatinine <=1.5 mg/dL or a calculated glomerular filtration rate >60 mL/min/1.73 m2 (Cockcroft-Gault) CPK <2.5 x upper limit of normal (ULN) Have total bilirubin <xULN. If total bilirubin is > 1,5xULN, measure direct and indirect bilirubin to evaluate for Gilbert's syndrome (if direct bilirubin is within normal range, subject may be eligible).  Have alkaline phosphatase (ALP) 2.5xULN; if the ALP is >2.5xULN, then an ALP liver fraction or 5' nucleotidase must be <ULN (as reported in absolute units of measure).  Have AST and ALT 2.5xULN. Have LVEF by MUGA scan or 2D-ECHO within normal limits for the institution.  Patients must be in postmenopausal (at least 12 months consecutive amenorrhea, in the appropriate age group and without other known or suspected cause), or have been sterilized surgically Adequate recovery from acute toxicity of any prior treatment  Exclusion criteria:  Non-epithelial ovarian or mixed epithelial/non epithelial tumors (e.g., Mullerian tumors) Patients who did not respond to last platinum-based therapy or in whom last relapse occurred < 6 months from the last dose of platinum Bowel obstruction, sub-occlusive disease or the presence of symptomatic brain metastases Known hypersensitivity to any of the components of pegylated liposomal doxorubicin or TRABECTEDIN i.v. formulation Previous treatment with Trabectedin Known hypersensitivity to dexamethasone Less than 4 weeks from last dose of therapy with any investigational agent, or chemotherapy. History of another neoplastic disease (except basal cell carcinoma or cervical carcinoma in situ adequately treated) unless in remission for 3 years or longer Myocardial failure within six months before enrolment, New York Association (NYHA) Class II or worse heart failure, uncontrolled angina, severe uncontrolled ventricular arythmias, clinically significant pericardial disease, or electrocardiographic evidence of acute ischemic or active conduction system abnormalities Known significant chronic liver disease, such as cirrhosis or active hepatitis (potential subjects who test positive for hepatitis B surface antigen or hepatitis C antibodies are allowed provided they do not have active disease requiring antiviral therapy). Also known history of liver carcinoma and cholangitis with abnormal liver functionality Concurrent severe medical problems or any unstable medical condition unrelated to malignancy, which would significantly limit full compliance with the study or expose the patient to extreme risk or decreased life expectancy Known clinically relevant CNS metastases Psychiatric disorder that prevents compliance with protocol Patients with concurrent serious or uncontrolled infection Patients in need of yellow fever vaccine while on study chemotherapy Active infection Breastfeeding women Pregnant women Any other unstable medical condition