Inclusion Criteria: - Locally advanced or spreading liposarcoma or leiomyosarcoma that has been diagnosed through tissue analysis and is unable to be removed by surgery - Treated with an anthracycline and ifosfamide administered either in combination or as sequential regimens - ¿Measurable disease at baseline in accordance with Response Evaluation Criteria In Solid Tumors (RECIST) Version 1.1 - ¿Pathology specimens (eg, tumor blocks or unstained slides) for potential centralized pathology review and biomarker studies - ¿Eastern Cooperative Oncology Group (ECOG) performance status score of 0 or 1 Exclusion Criteria: - Prior exposure to trabectedin or dacarbazine - Less than 3 weeks from last dose of systemic cytotoxic therapy, radiation therapy, or therapy with any investigational agent - Other malignancy within past 3 years (exceptions: basal or nonmetastatic squamous cell carcinoma of the skin, cervical carcinoma in situ, or Federation Internationale de Gynecologie et d'Obstetrique (FIGO) Stage 1 carcinoma of the cervix) - Known central nervous system metastasis - Active liver disease, such as chronic viral hepatitis or cirrhosis - Heart attack within 6 months before enrollment - Any condition that, in the opinion of the investigator, would compromise the well-being of the patient or the study or prevent the patient from meeting or performing study requirements