Inclusion Criteria:  For Enrollment to Phase 1b: Participants must have histologically confirmed advanced or metastatic soft-tissue sarcoma and no curative multimodality treatment options available. For Enrollment to Phase 2: Participants must have histologically confirmed advanced or metastatic leiomyosarcoma (LMS) and no curative multimodality treatment options available. Participants must have measurable disease per RECIST 1.1 criteria Age ≥ 18 years. Because no dosing or adverse event data are currently available on the use of lurbinectedin in combination with doxorubicin in participants <18 years of age, children are excluded from this study, but will be eligible for future pediatric trials. ECOG performance status ≤ 2 (Karnofsky ≥ 60%,).  Participants must have adequate organ and marrow function as defined below:  Absolute Neutrophil Count ≥ 1,500/mcL Hemoglobin (Hgb) ≥ 8 g/dl (transfusion support permitted) Platelet Count ≥ 100,000/mcL Total Bilirubin ≤ 1.5 × institutional upper limit of normal (ULN) AST (SGOT) / ALT(SGPT) ≤ 2.5 × institutional ULN, OR ≤ 5 × institutional ULN if elevation is a result of metastases Creatinine ≤ 1.5 × institutional ULN, OR Creatinine Clearance ≥ 60 mL/min/1.73 m2 for participants with creatinine levels above 1.5 × institutional normal (calculated via the Cockcroft-Gault equation) Creatine Phosphokinase (CPK)< 2.5 × institutional ULN on two different determinations performed one week (± 1 day) apart For participants with known chronic hepatitis B virus (HBV) infection, the HBV viral load must be undetectable on suppressive therapy, if indicated. Participants with a known history of hepatitis C virus (HCV) infection must have been treated and cured. For participants with known HCV infection who are currently on treatment, they are eligible if they have an undetectable HCV viral load. Left ventricular ejection fraction (LVEF) ≥ 50% on screening echocardiogram (ECHO) or multigated acquisition (MUGA) scan. The effects of lurbinectedin or doxorubicin on the developing human fetus are unknown. For this reason and because anti-cancer agents are known to be teratogenic, women of child-bearing potential and men must agree to use adequate contraception (hormonal or barrier method of birth control; abstinence) prior to study entry and for the duration of study participation. Should a woman become pregnant or suspect she is pregnant while she or her partner is participating in this study, she should inform her treating physician immediately. Men and women treated or enrolled on this protocol must agree to use adequate contraception prior to the study, for the duration of study participation, and 6 months after completion of study agent administration. Ability to understand and the willingness to sign a written informed consent document. Participants must have archival tissue available for analysis in the form of a formalin-fixed paraffin embedded (FFPE) block or unstained slides. Participants without archival tissue available may be enrolled with approval of the Sponsor-Investigator. Note: confirmation of availability of archival tissue is the only requirement for eligibility, archival tissue does not need to be received by the study team or site prior to enrollment. Patients with a prior or concurrent malignancy whose natural history or treatment does not have the potential to interfere with the safety or efficacy assessment of the investigational regimen as assessed by the treating investigator may be included with the approval of the Sponsor-Investigator.  Exclusion Criteria:  Participants who have received prior anthracycline or trabectedin (Yondelis, ET-743), including prior exposure to doxorubicin or liposomal doxorubicin. Participants who have received more than 2 prior lines of cytotoxic chemotherapy for the phase 1b study and no more than 1 prior line of cytotoxic chemotherapy for the phase 2 study. There is no limit on the number of prior lines of non-cytotoxic chemotherapy (e.g., pazopanib, immunotherapy). Prior exposure to lurbinectedin (PM01183). Participants who have received prior radiation treatment of > 45 Gy to the pelvis. Participants who have received or undergone prior chemotherapy within 14 days of cycle 1 day 1, therapeutic radiation therapy within 21 days of cycle 1 day 1 or major surgery within 21 days of cycle 1 day 1. Participants who have received prior palliative radiation therapy within 7 days of cycle 1 day 1. Participants who have received prior antibody-based therapy (e.g., nivolumab) within 4 weeks or 3 half-lives (whichever is shorter) of cycle 1 day 1. Participants who have received prior oral small molecule or tyrosine kinase inhibitor (TKI) therapy within 2 weeks or 3 half-lives (whichever is shorter) of cycle 1 day 1. Participants who have not recovered to ≤ Grade 1 or baseline from adverse events attributed to any prior anti-cancer therapy, with the exceptions of alopecia, controlled endocrine toxicity (e.g., hypothyroidism), and cutaneous toxicity which will be permitted at Grade 2. Participants who are receiving any other investigational agents. Participants with known CNS disease involvement, with the exception of patients with brain metastases that have been previously treated and have remained stable on MRI ≥ 28 days prior to cycle 1 day 1 without use of steroids or anti-epileptic medications. History of allergic reactions attributed to compounds of similar chemical or biologic composition to lurbinectedin or doxorubicin. Participants receiving any medications or substances that are strong or moderate inhibitors or inducers of CYP3A, CYP2D6, or P-gp are ineligible. Because the lists of these agents are constantly changing, it is important to regularly consult a frequently updated medical reference. As part of the enrollment/informed consent procedures, the participant must be counseled on the risk of interactions with other agents, and what to do if new medications need to be prescribed or if the participant is considering a new over-the-counter medicine or herbal product. Uncontrolled intercurrent illness including, but not limited to: ongoing or active infection, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia, chronic indwelling drains, or psychiatric illness/social situations that would limit compliance with study requirements. History of interstitial pneumonitis or pulmonary fibrosis. Known cardiomyopathy. Pregnant women are excluded from this study because lurbinectedin and doxorubicin are anti-cancer agents with the potential for teratogenic or abortifacient effects. Because there is an unknown but potential risk for adverse events in nursing infants secondary to treatment of the mother with lurbinectedin or doxorubicin, breastfeeding must be discontinued if the mother is treated with lurbinectedin or doxorubicin. A negative pregnancy test is required for women of childbearing potential prior to the first dose of study medication. Immunocompromised patients, including patients who are known to be seropositive for human immunodeficiency virus (HIV) due to the increased risk of lethal infections when treated with marrow-suppressive therapy. HIV testing is not required as part of screening.