Copanlisib in Combination With Romidepsin in Patients With Relapsed or Refractory Mature T-cell Lymphoma

Inclusion Criteria: * Be willing and able to provide written informed consent for the trial. * Be ≥18 years of age on day of signing informed consent. * Have measurable disease based on the Lugano Criteria. * Phase I: patient must have histologically confirmed R/R NHL or HL (defined by World Health Organization (WHO) criteria). * Expansion phase: patients must have histologically confirmed R/R mature T-cell lymphoma (defined by WHO criteria). * Patient must have received at least two prior lines of therapy prior to enrollment in this study. * Have a performance status of 0-2 on the Eastern Cooperative Oncology Group (ECOG) Performance Scale. * Left Ventricular Ejection Fraction (LVEF) \> 50%. * Demonstrate adequate organ function. All screening labs should be performed within 10 days of treatment initiation. The following treatments are prohibited: (a) Chemotherapy, monoclonal antibody within 4 weeks; (b) radiotherapy within 2 weeks prior to entering the study; (c) systemic steroids that have not been stabilized (≥ 5 days) to the equivalent of ≤10 mg/day prednisone prior to the start of the study drugs; (d) other concurrent investigational agents within 4 weeks prior to entering the study; (e) use of copanlisib or romidepsin within the past 3 months. * Patients that have not recovered from adverse events due to chemotherapy agents administered more than 4 weeks earlier. * Hypersensitivity to copanlisib or romidepsin or any of its excipients. * Patients that received major surgery and have not recovered adequately from the toxicity and/or complications from the intervention prior to starting therapy. * Patients with active, clinically serious infections \> CTCAE Grade 2. * Patients with Cytomegalovirus (CMV) PCR positive at baseline. * Patients with active infections including: active tuberculosis (TB), Hepatitis B (e.g., HBsAg reactive), Hepatitis C, Hepatitis A. * Patients with uncontrolled inter-current illness (e.g. pneumonia). * Patients with known history of Human Immunodeficiency Virus (HIV) (HIV 1/2 antibodies * Adequate Contraception. Exclusion Criteria: * The following treatments are prohibited: (a) Chemotherapy, monoclonal antibody within 4 weeks; (b) radiotherapy within 2 weeks prior to entering the study; (c) systemic steroids that have not been stabilized (≥ 5 days) to the equivalent of ≤10 mg/day prednisone prior to the start of the study drugs; (d) other concurrent investigational agents within 4 weeks prior to entering the study; (e) use of copanlisib or romidepsin within the past 3 months. * Patients that have not recovered from adverse events due to chemotherapy agents administered more than 4 weeks earlier. * Hypersensitivity to copanlisib or romidepsin or any of its excipients. * Patients that received major surgery and have not recovered adequately from the toxicity and/or complications from the intervention prior to starting therapy. * Patients with active, clinically serious infections \> CTCAE Grade 2. * Patients with CMV PCR positive at baseline. * Patients with active infections including: active TB (Mycobacterium Tuberculosis), Hepatitis B (e.g., HBsAg reactive), Hepatitis C (e.g., HCV RNA \[qualitative\] is detected), Hepatitis A. * Patients with uncontrolled inter-current illness (e.g. pneumonia). * Patients with known history of Human Immunodeficiency Virus (HIV) (HIV 1/2 antibodies The following treatments are prohibited: (a) Chemotherapy, monoclonal antibody within 4 weeks; (b) radiotherapy within 2 weeks prior to entering the study; (c) systemic steroids that have not been stabilized (≥ 5 days) to the equivalent of ≤10 mg/day prednisone prior to the start of the study drugs; (d) other concurrent investigational agents within 4 weeks prior to entering the study; (e) use of copanlisib or romidepsin within the past 3 months. * Patients that have not recovered from adverse events due to chemotherapy agents administered more than 4 weeks earlier. * Hypersensitivity to copanlisib or romidepsin or any of its excipients. * Patients that received major surgery and have not recovered adequately from the toxicity and/or complications from the intervention prior to starting therapy. * Patients with active, clinically serious infections \> CTCAE Grade 2. * Patients with CMV PCR positive at baseline. * Patients with active infections including: active TB (Mycobacterium Tuberculosis), Hepatitis B (e.g., HBsAg reactive), Hepatitis C (e.g., HCV RNA \[qualitative\] is detected), Hepatitis A. * Patients with uncontrolled inter-current illness (e.g. pneumonia). * Patients with known history of Human Immunodeficiency Virus (HIV) (HIV 1/2 antibodies * Patients with known active concurrent malignancy (except non-melanoma skin cancer, prostatic intraepithelial neoplasia, or carcinoma in situ of the cervix). If there is a history of prior malignancy, the patient must be disease-free for ≥ 3 years. * Patients with known active central nervous system (CNS) metastases and/or carcinomatous meningitis and/or known lymphomatous involvement of the central nervous system. * Evidence of active non-infectious pneumonitis (NIP). * History of concurrent condition of interstitial lung disease of any severity and/or severely impaired lung function (as judged by the investigator). * Patients with an active autoimmune disease that has required systemic treatment in the past 2 years (i.e. with use of disease modifying agents, corticosteroids or immunosuppressive drugs). Of note, replacement therapy (e.g. thyroxine, insulin, or physiologic corticosteroid replacement therapy for adrenal or pituitary insufficiency, etc.) is allowed for patient on study. * Uncontrolled Type I or II diabetes mellitus as deemed appropriate by the investigator (suggested guidelines for uncontrolled diabetes: HbA1c\> 8.5%). * Uncontrolled hypertension, i.e., blood pressure (BP) of ≥ 150/90; patients who have a history of hypertension controlled by medication must be on a stable dose (for at least one month) and meet all other inclusion criteria. * Concomitant use of CYP3A4 inhibitors. * Congenital long QT syndrome and/or QTc interval ≥ 500 milliseconds. * Patients taking drugs leading to significant QT prolongation. * Any cardiac arrhythmia requiring an anti-arrhythmic medication, with the exception of stable doses of beta-blockers. * Myocardial infarction within 6 months of cycle 1, day 1. \[Subjects with a history of myocardial infarction between 6 and 12 months prior to cycle 1, day 1, who are asymptomatic and have had a negative cardiac risk assessment (treadmill stress test, nuclear medicine stress test, or stress echocardiogram) since the event, may participate\]. * Symptomatic coronary artery disease (CAD), e.g., angina Canadian Class II-IV. In any patient in whom there is doubt, the patient should have a stress imaging study and, if abnormal, angiography to define whether or not CAD is present. * An ECG recorded at screening showing evidence of cardiac ischemia (ST depression of ≥ 2 mm, measured from isoelectric line to the ST segment). If in any doubt, the patient should have a stress imaging study and, if abnormal, angiography to define whether or not CAD is present. * Congestive heart failure (CHF) that meets New York Heart Association (NYHA) Class II to IV definitions and/or ejection fraction \< 40% by multigated acquisition (MUGA) scan or \< 50% by echocardiogram and/or magnetic resonance imaging (MRI); * Hypertrophic cardiomegaly or restrictive cardiomyopathy from prior treatment or other causes. * Arterial or venous thrombotic or embolic events such as cerebrovascular accident (including transient ischemic attacks), deep vein thrombosis or pulmonary embolism within 3 months before the start of study medication. * Patients requiring dual anti-platelets treatment for cardiac conditions or patients who are on anticoagulation for arterial or venous thrombosis. Patients with known psychiatric or substance abuse disorders that would interfere with cooperation with the requirements of the trial. * Patients that are pregnant or breastfeeding, or expecting to conceive or father children within the projected duration of the trial, starting with the pre-screening or screening visit through at least 30 days after the last dose of trial treatment.