Inclusion Criteria:

• Age ≥ 16 years.
• Advanced unresectable or metastatic sarcoma.

Cohort 1

• Leiomyosarcoma (LMS)/Liposarcoma (LPS).

Cohort 2

• Other sarcoma histologies (excluding gastrointestinal stromal tumors).
• Received at least 1 prior standard chemotherapy. For cohort 1 patients, this must have included a prior anthracycline.
• Measurable disease by RECIST 1.1.
• Adequate hematologic, renal, hepatic function.
• Adequate creatine phosphokinase.
• ECOG performance status ≤ 1.
• Left ventricular ejection fraction (LVEF) ≥ institutional lower limit of normal (LLN).
• Women of childbearing potential and men must agree to use adequate contraception from signing informed consent to at least 6 months (females) and 5 months (men) after study drug treatment.
• Men must agree to use barrier contraception or abstinence and not donate sperm during treatment and for 5 months after the last dose of study treatment. Female partners of male patients should also use a highly effective form of contraception if they are of childbearing potential.
• Left ventricular ejection fraction ≥ institutional LLN.
• Ability to understand and the willingness to sign a written informed consent.
• Must have a life expectancy ≥ 16 weeks.

Exclusion Criteria:

• Prior therapy with PARP inhibitor, including olaparib.
• Prior therapy with trabectedin.
• Additional active malignancy or treatment for alternative cancer (excluding non-melanoma skin cancer) requiring treatment within the past two years.
• Pregnant or breastfeeding women.
• Known hypersensitivity to trabectedin or olaparib.
• Patients with symptomatic uncontrolled brain metastases. A scan to confirm the absence of brain metastases is not required. The patient can receive a stable dose of corticosteroids before and during the study as long as these were started at least 4 weeks prior to treatment. Patients with spinal cord compression unless considered to have received definitive treatment for this and evidence of clinically stable disease for 28 days.
• Patients considered a poor medical risk due to a serious, uncontrolled medical disorder, non-malignant systemic disease or active, uncontrolled infection. Examples include, but are not limited to, uncontrolled ventricular arrhythmia, recent (within 3 months) myocardial infarction, uncontrolled major seizure disorder, unstable spinal cord compression, superior vena cava syndrome, extensive interstitial bilateral lung disease on High Resolution Computed Tomography (HRCT) scan or any psychiatric disorder that prohibits obtaining informed consent.
• Resting ECG indicating uncontrolled, potentially reversible cardiac conditions, as judged by the investigator (eg., unstable ischemia, uncontrolled symptomatic arrhythmia, congestive heart failure, QTcF prolongation > 500 ms, electrolyte disturbances, etc.), or patients with congenital long QT syndrome.
• Persistent toxicities (>Common Terminology Criteria for Adverse Event (CTCAE) grade 2) caused by previous cancer therapy, excluding alopecia.
• Patients with myelodysplastic syndrome (MDS) /acute myeloid leukemia (AML) or with features suggestive of MDS/AML.
• Patients unable to swallow orally administered medication and patients with gastrointestinal disorders likely to interfere with absorption of the study medication.
• Immunocompromised patients; e.g., patients who are known to be serologically positive for human immunodeficiency virus (HIV).
• Patients with known active hepatitis (i.e., Hepatitis B or C).
• Patients receiving any systemic chemotherapy or radiotherapy (except for palliative reasons) within 3 weeks prior to study treatment.
• Concomitant use of known strong CYP3A inhibitors (e.g., itraconazole, telithromycin, clarithromycin, protease inhibitors boosted with ritonavir or cobicistat, indinavir, saquinavir, nelfinavir, boceprevir, telaprevir) or moderate CYP3A inhibitors (e.g, ciprofloxacin, erythromycin, diltiazem, fluconazole, verapamil). The required washout period prior to starting study treatment is 2 weeks.
• Concomitant use of known strong (e.g., phenobarbital, enzalutamide, phenytoin, rifampicin, rifabutin, rifapentine, carbamazepine, nevirapine and St. John’s Wort) or moderate CYP3A inducers (eg. bosentan, efavirenz, modafinil). The required washout period prior to starting study treatment is 5 weeks for enzalutamide or phenobarbital and 3 weeks for other agents.
• Major surgery within 2 weeks of starting study treatment and patients must have recovered from any effects of any major surgery.
• Participation in another clinical study with an investigational product administered in the last 30 days prior to anticipated study treatment.
• Judgment by the investigator that the patient should not participate in the study if the patient is unlikely to comply with study procedures, restrictions and requirements.
• Has received a live vaccination with 2 weeks of enrollment.
• Previous allogeneic bone marrow transplant or double umbilical cord blood transplantation.
• Involvement in the planning and/or conduct of the study.
• Previous enrollment in the present study.