A Study of Alectinib and Duvelisib in People With Anaplastic Lymphoma Kinase-Positive Anaplastic Large Cell Lymphoma (ALK+ALCL)

Inclusion Criteria: * Pathologically-confirmed diagnosis of ALK+ ALCL by WHO/ICC, classification procedures in use at the time of diagnosis. Note that ALK+ ALCL by definition expresses ALK, which is readily detectable on standard IHC. Confirmation through molecular sequencing of the specific ALK translocation and fusion partner is not necessary for enrollment. * Relapsed or refractory disease after at least one line of prior systemic therapy. * NOTE: Prior systemic therapy must have included at least one cytotoxic chemotherapy agent. * NOTE: Prior treatment with an ALK inhibitor is allowed. * NOTE: Patients being treated with an ALK inhibitor immediately prior to enrollment are eligible. This includes patients on an ALK inhibitor who are in clinical remission at the time of enrollment, as long as the patient is not immediately planned for allogeneic transplant. * NOTE: If the last therapy was an ALK inhibitor, the patient must not have stopped the ALK inhibitor and maintained clinical remission (no relapse) with no intervening therapy for ≥ six months. * NOTE: Prior progression on ALK inhibitor is not specifically exclusionary though should be reviewed with the MSK Principal Investigator. * Age ≥ 18 years at the time of enrollment * ECOG performance status ≤ 2 at the time of enrollment. * Laboratory criteria: * Absolute neutrophil count ≥ 1.0 K/mcL or ≥ 0.5 K/mcL if due to lymphoma (NOTE: growth factor is allowed). * Platelet count ≥ 75 K/uL. * Calculated creatinine clearance ≥ 60 mL/min by Cockcroft-Gault. * Total bilirubin ≤ 2x upper limit of normal (ULN) or ≤ 3x ULN if due to hepatobiliary involvement with lymphoma, or ≤ 3x ULN if history of Gilbert's disease. * Aspartate (AST) and alanine (ALT) aminotransferase ≤ 3 x ULN or ≤ 5x ULN if due to hepatobiliary involvement with lymphoma. * NOTE: Patients with AST and/or ALT \> 3x ULN and total bilirubin \> 2x ULN must be reviewed with the MSK Principal Investigator to determine eligibility. * NOTE: Patients must meet laboratory criteria prior to initiation of the alectinib lead-in cycle and prior to initiation of combination therapy with duvelisib * Able to swallow pills. * Able to take prophylactic medications against Pneumocystis jirovecii pneumonia (PJP) * Women of reproductive potential must have a negative serum or urine β human chorionic gonadotropin (β-HCG) pregnancy test within 14 days before initiating therapy. * Females of childbearing age must be on effective contraception per institutional standards during the treatment period and for 5 weeks after the last dose of the study drugs. * Males must consistently use an effective contraception method per institutional standards during the treatment period and for 3 months following the last dose of the study drugs. Exclusion Criteria: * Prior allogeneic stem cell transplant within 6 months of starting treatment or patients with active graft versus host disease (GVHD). * Previous systemic anti-cancer therapy for ALK+ ALCL within 7 days of initiating study drug * NOTE: Systemic corticosteroids are allowed and must be tapered to 10 mg/day or less (prednisone equivalent) upon start of investigational treatment. * NOTE: Patients who have received localized radiotherapy as part of immediate prior therapy may be allowed to enroll with shorter washout period after discussion with the MSK Principal Investigator. * NOTE: Prior progression on ALK inhibitor is not specifically exclusionary though should be reviewed with the MSK Principal Investigatory. * Ongoing use of immunosuppressant medications, including corticosteroids greater than 10 mg of prednisone or equivalent at the time of enrollment. * Prior gastrointestinal condition or surgery that may, in the investigator's judgment, adversely affect drug absorption. * Active viral infection with hepatitis B or hepatitis C. For hepatitis B, patients who are seropositive (hepatitis B core Ab positive) are permitted if HBV DNA is negative by PCR. For hepatitis C, patients who are seropositive (hepatitis C Ab positive) are eligible if HCV DNA is negative by PCR and curative therapy has been completed. o NOTE: Patients with HIV infection are permitted to enroll but are required to be on antiretroviral regimens that are in accordance with the current International AIDS Society guidelines on concurrent use with chemotherapy. Use of experimental antiretroviral agents or those containing zidovudine or ritonavir, cobicistat or similar potent CYP3 inhibitors are prohibited. In order to be eligible, patients taking zidovudine or ritonavir, or cobicistat or other CYP3 inhibitors must change to a different regimen 7 days prior to therapy initiation. Subjects must be on HAART for at least 12 weeks prior to therapy. * Concurrent malignancy requiring active therapy within the last 2 years with the exception of basal cell or squamous cell carcinoma limited to the skin, carcinoma in situ of the cervix, breast or localized prostate cancer. Adjuvant or maintenance therapy to reduce the risk of recurrence or other malignancy is permissible after discussion with the Principal Investigator. * Active cytomegalovirus (CMV) as defined by positive CMV PCR with clinical manifestations consistent with active CMV infection and requiring therapy. Carriers will be managed as per institutional guidelines. * Patients should not be on CYP4503A inhibitors or inducers at the time of treatment initiation. * Pregnant or breastfeeding women. * Any serious or unstable medical condition, laboratory abnormality, or psychiatric illness that would prevent the patient from signing informed consent or, in the investigator's judgment, increase the risk to the patient associated with participation in the study