INCLUSION CRITERIA:  Subjects must have a histologic diagnosis, confirmed by the Laboratory of Pathology, NCI with one of the following subtypes and prior therapy, as follows:  At least 1 anthracycline-containing regimen:  Burkitt lymphoma Burkitt-like lymphoma with 11q aberration High-grade B-cell lymphoma, NOS High-grade B-cell lymphoma, with MYC and BCL2 and/or BCL6 rearrangements  OR  --Must have had at least 2 prior regimens, 1 of which must have been anthracyclinecontaining regimen OR be primary refractory to frontline therapy:  ---DLBCL, NOS, Germinal center B-cell type (GCB) type;  NOTE: subjects with coexisting or a history of indolent lymphoma are eligible (i.e., transformed lymphoma )  ---T-cell/histocyte-rich large B-cell lymphoma  Measurable or evaluable disease on imaging scans or bone marrow No other current systemic anti-lymphoma therapy. NOTE: Recent short-term (less than or equal to 7 days) use of corticosteroids or prior radiation to sites of disease involvement is permitted. Any HIV status will be included in this study as long as infection is controlled; in the opinion of the investigator. Status must be confirmed at screening and the subject must be willing to take any recommended antiretroviral therapy. Greater than or equal to 18 years of age on day of signing informed consent ECOG performance status less than or equal to 2  Adequate organ function as evidenced by the following laboratory parameters, unless dysfunction is secondary to lymphoma involvement as determined by the investigator:  Absolute neutrophil count (ANC) greater than or equal to 1,000 /mm3 Platelets greater than or equal to 75 x 109 /L Hemoglobin greater than or equal to 8 g/dL (unless due to disease itself, transfusion permitted to meet criteria) Renal function Glomerular filtration rate (GFR) >40ml/min/1.73 m2 as estimated by Modification of Diet in Renal Disease  (MDRD) abbreviated formula. If not on target, a 24- hour urine creatinine clearance can be used to directly measure.  Total bilirubin less than or equal to 1.5 x ULN OR < 1.5-3.0 x ULN for subjects with liver involvement*  AST and ALT less than or equal to 3.0 x ULN OR < 5 x ULN for subjects with liver involvement  Acceptable range as long as there is no persistent nausea, vomiting, right upper quadrant pain or tenderness, fever, rash, or eosinophilia  Must have fully recovered from all effects of prior surgery. NOTE: Minor procedures requiring Twilight sedation, such as tissue biopsies, endoscopies or mediport placement require a 24-hour waiting period prior to treatment initiation. Women of childbearing potential (WOCBP) and men must agree to use effective contraception when sexually active. This applies for the time period between signing of the informed consent form and for at least 3 months after the last dose of copanlisib and 12 months after the last dose of rituximab, whichever is later. Women of childbearing potential must have a serum pregnancy test performed a maximum of 7 days before start of treatment, and a negative result must be documented before start of treatment.  NOTE: A woman is considered of childbearing potential, (i.e., fertile), following menarche and until becoming post-menopausal unless permanently sterile. Permanent sterilization methods include but are not limited to hysterectomy, bilateral salpingectomy and bilateral oophorectomy. A postmenopausal state is defined as no menses for continuous 12 months without an alternative medical cause. A high follicle stimulating hormone (FSH) level in the postmenopausal range may be used to confirm a postmenopausal state in women not using hormonal contraception or hormonal replacement therapy. The investigator or a designated associate is requested to advise the subject how to achieve highly effective birth control (failure rate of less than 1%), e.g., intrauterine device (IUD), intrauterine hormone-releasing system (IUS), bilateral tubal occlusion, vasectomized partner and sexual abstinence. The use of condoms by male subjects is required unless the female partner is permanently sterile.  Willingness to have a central venous access line placed if the subject does not already have one in place Ability of patient to understand and the willingness to sign a written informed consent document  EXCLUSION CRITERIA:  Subjects previously exposed to, intolerant of, or ineligible for PI3K inhibitors and/or their combination Brain parenchymal involvement CMV-positive PCR at screening History of diabetic ketoacidosis  Uncontrolled intercurrent illness including, but not limited to the following that may limit interpretation of results or that could increase risk to the subject at the discretion of the investigator:  Any secondary malignancy that requires active systemic therapy Diabetes mellitus with Hgb A1C > 8.5 Clinically significant interstitial lung disease and/or lung disease that severly impairs lung function Uncontrolled HIV Active hepatitis C infection. NOTE: Subjects who are hepatitis C antibody positive will need to have a negative HCV PCR result before enrollment. Those with a positive PCR for hepatitis C are excluded. Active hepatitis B infection. NOTE: Patients who are hepatitis B surface antigen (HbcsAg) or hepatitis B core antibody (HbcAb) positive will need to have a negative HBV PCR result before enrollment. Those with a positive PCR for hepatitis B are excluded. Those who are hepatitis B surface antigen (HbcsAg) or hepatitis B core antibody (HbcAb) positive with a negative PCR for hepatitis B will be treated with antivirals designed to prevent hepatitis B reactivation (e.g., entecavir) throughout therapy and for 12 months after therapy and have monitoring for hepatitis B reactivation with PCR.  Clinically significant, uncontrolled heart disease and/or cardiac repolarization abnormalities, including any of the following:  History of acute coronary syndromes (including myocardial infarction, unstable angina, coronary artery bypass grafting, coronary angioplasty, or stenting) or symptomatic pericarditis within 6 months prior to screening Congestive heart failure (New York Heart Association functional classification III-IV) Unstable angina Left Ventricular Ejection Fraction (LVEF) <40% as determined by echocardiogram (ECHO) at screening Clinically significant cardiac arrhythmias (e.g., ventricular tachycardia), complete left bundle branch block, high-grade AV block (e.g., bifascicular block, Mobitz type II and third-degree AV block)  Long QT syndrome or family history of idiopathic sudden death or congenital long QT syndrome, or any of the following:  Long QT syndrome or family history of idiopathic sudden death or congenital long QT syndrome, or any of the following: Inability to determine the QT interval on screening (QTcF, using Fredericia s correction) Long QT syndrome or family history of idiopathic sudden death or congenital long QT syndrome Known mental or physical illness that would interfere with cooperation with the requirements of the trial or confound the results or interpretation of the results of the trial and, in the opinion of the treating investigator, would make the subject inappropriate for entry into the study. Requirement to continue on any of the medications that are excluded Breast-feeding subjects. Because there is an unknown but potential risk for adverse events in nursing infants secondary to treatment of the mother with these agents, breastfeeding should be discontinued if the mother is treated on study.