A Phase II Multi-Strata Study of PM01183 as a Single Agent or in Combination With Conventional Chemotherapy in Metastatic and/or Unresectable Sarcomas

Inclusion Criteria: * Participants must have pathologically confirmed soft-tissue sarcoma, which is metastatic or unresectable, sarcoma with no curative multimodality options * Participants must have measurable disease by Response Evaluation Criteria in Solid Tumors (RECIST) 1.1. * No more than two prior lines of chemotherapy for metastatic sarcoma are allowed; Neo-adjuvant/adjuvant chemotherapy with definitive therapy (radiation, surgery or radiation and surgery) will not be counted as one of these prior lines of therapy. * Age ≥ 18 and ≤ 75 years. * Eastern Cooperative Oncology Group performance status ≤1 (see Appendix A) * Life expectancy of greater than 3 months * Participants must have normal organ and marrow function as defined below: * Hemoglobin ≥ 9 g/dl * absolute neutrophil count ≥ 1,500/mcL * platelets ≥ 100,000/mcL * total bilirubin ≤ 1.5 X ULN * AST(SGOT)/ALT(SGPT) ≤3 X ULN (including patients with liver metastases) * creatinine ≤1.5 X ULN * CPK \< 2.5 X ULN * Albumin ≥ 3 g/dl * Women of childbearing potential must have a negative serum or urine pregnancy test within 7 days prior to receiving study agents. * For patients in stratum A, an echocardiogram or multiple gated acquisition scan (MUGA) demonstrating left ventricular ejection fraction \> 50% is required within 30 days prior to study drug administration. * Participants must be willing and able to comply with the study scheduled visits, laboratory tests, and other procedures outlined in the protocol. * Pre-menopausal women must have a negative pregnancy test before study entry. Both women and men must agree to use a medically acceptable method of contraception throughout the treatment period and for at least six weeks after treatment discontinuation. Acceptable methods of contraception include intrauterine device (IUD), oral contraceptive, subdermal implant, double barrier and/or complete abstinence (non-periodic). * Washout period prior to Day 1 Cycle 1: * ≥ 3 weeks since last chemotherapy or therapeutic radiation therapy (RT) * ≥ 4 weeks or 3 half-lives since prior antibody-based therapy, whichever is shorter * ≥ 2 weeks since any oral anti-neoplastic or oral investigational agent * Resolution of treatment-related toxicity to ≤ grade 1; alopecia and cutaneous toxicity are allowed ≤ grade 2. * ≥1 week since palliative RT * Ability to understand and the willingness to sign a written informed consent document. Exclusion Criteria: * Prior exposure to PM01183 * Patients who have received trabectedin (Yondelis, ET-743) or participated in the phase III clinical study of trabectedin NCT01343277 previously will not be eligible. * For stratum A, patients must not have received prior anthracycline-based therapy (prior treatment with non-anthracyclines is permitted). * For stratum B patients must have received prior anthracycline-based therapy (or have a contraindication to receiving this treatment) and must not have received prior gemcitabine * For stratum C, patients must have received prior anthracycline or gemcitabine-based therapy, or had a contraindication to either or both * Prior radiation treatment of \>45 Gy to the pelvis * Previously untreated Ewing Sarcoma and rhabdomyosarcoma * Non-soft tissue sarcomas, such as osteosarcoma and chondrosarcoma are excluded * Participants who are receiving any other investigational agents. * Active hepatopathy of any origin including active hepatitis B and hepatitis C * Participants with known uncontrolled brain metastases will be excluded from this clinical. * History of allergic reactions attributed to compounds of similar chemical or biologic composition to PM01183 or trabectedin (Yondelis, ET-743). * Uncontrolled intercurrent illness including, but not limited to, ongoing or active infection, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia, chronic indwelling drains, history of interstitial pneumonitis or pulmonary fibrosis or psychiatric illness/social situations that would limit compliance with study requirements. * Actively breastfeeding women unless it is interrupted during treatment and at least 6 weeks after treatment discontinuation. * Known myopathy or persistent CPK elevations \>2.5 ULN in two different determinations performed one week apart. * Immunocompromised patients, including those with HIV.