Patients with known human immunodeficiency virus (HIV) positive must have a cluster of differentiation (CD)4 cell count be >= 350 cells/mm^3 within 14 days prior to study entry (note, however, that HIV testing is not required for entry into this protocol) Patients with active human immunodeficiency virus (HIV) infection are eligible if their cluster of differentiation (CD)4 count is > 499/cu mm and their viral load is < 50 copies/ml; use of highly active antiretroviral treatment (HAART) is allowed Human immunodeficiency virus (HIV)-positive patients on combination antiretroviral therapy will be eligible unless the cluster of differentiation (CD)4 count is < 200 cells/mm^3 within one month of study enrollment Human immunodeficiency virus (HIV)-positive patients who are not receiving: agents with the potential for PK interactions with romidepsin or hepatotoxic antiretrovirals (nucleoside reverse-transcriptase inhibitors [NRTIs]: abacavir, didanosine, emtricitabine, lamivudine, stavudine, and zidovudine), dual protease inhibitor (PI)-based regimens except low-dose boosting with ritonavir, atazanavir, indinavir, maraviroc, and nevirapine may be eligible; additionally, the HIV-positive patients should have a cluster of differentiation (CD)4 count > 250/mm^3; if the specific cause of hepatic dysfunction is unknown, the patient should be worked up for other viral causes of hepatitis and their eligibility determined after consultation with the principal investigator Patients who are known to be human immunodeficiency virus positive (HIV+) are eligible providing they meet all of the following additional criteria within 28 days prior to registration:* Cluster of differentiation (CD)4 cells >= 500/mm^3* Viral load of < 50 copies HIV messenger ribonucleic acid (mRNA)/mm^3 if on combination antiretroviral therapy (cART) or < 25,000 copies HIV mRNA if not on cART* No zidovudine or stavudine as part of cART* Patients who are HIV+ and do not meet all of these criteria are not eligible for this study STEP I: Human immunodeficiency virus (HIV) infection is not excluded; known HIV positive patients must meet the following criteria:* Cluster of differentiation (CD)4 cell count >= 350/mm^3* No history of acquired immune deficiency syndrome (AIDS)-related illness* Not currently prescribed zidovudine or stavudine Immunocompromised patients (other than that related to the use of corticosteroids) with the exception of patients known to be human immunodeficiency virus (HIV) positive and have a cluster of differentiation 4 (CD4) count > 400 and do not require antiretroviral therapy A cluster of differentiation (CD)4+ lymphocyte count > 50/mcL will be required within 2 weeks of study participation Patients with human immunodeficiency virus (HIV) are eligible if they are not on antiviral agents and have adequate cluster of differentiation (CD)4 counts (>= 500 mm^3) Any evidence of severe or uncontrolled systemic disease, active infection, active bleeding diatheses or renal transplant, including any patient known to have hepatitis B, hepatitis C or human immunodeficiency virus (HIV) will be excluded; patients with HIV who have adequate cluster of differentiation (CD)4 count, not requiring antiretroviral medication will not be excluded Patients with human immunodeficiency virus (HIV) infection are eligible, provided they meet the following:* No history of acquired immunodeficiency syndrome (AIDS)-defining conditions or other HIV related illness* Cluster of differentiation (CD)4+ cells nadirs > 350/mm^3 within 28 days prior to registration* Treatment sensitive HIV and, if on anti-HIV therapy, HIV viral load < 50 copies/mm^3 within 28 days prior to registration* Note: HIV+ patients who enroll on this study and are assigned to treatment with ixazomib may need to modify their anti-retroviral therapy prior to receiving protocol therapy if they are on strong inducers or potent inhibitors of cytochrome P450 3A4 Human immunodeficiency virus (HIV)-positive patients on antiviral drugs and/or cluster of differentiation (CD)4 count is inadequate (< 500); if neither condition exists, HIV-positive patients are eligible Disease must be cluster of differentiation (CD)30 positive Patients with known human immunodeficiency virus (HIV) infection are eligible if they meet all of the following criteria:* No history of acquired immune deficiency syndrome (AIDS)-related complications other than a history of low CD4+ T-cell count (< 200/mm^3) prior to initiation of combination antiretroviral therapy; on study CD4+ T-cell count may not be informative due to leukemia and should not be used as an exclusion criterion if low* Patient must be healthy on the basis of HIV disease with high likelihood of near normal life span were it not for the leukemia* Patient must have serum HIV viral load of < 200 copies/mm^3* Patient must be on combination antiretroviral therapy with minimal pharmacokinetic interactions with study therapy and minimal overlapping clinical toxicity with protocol therapy* Patient must not be receiving protease inhibitors or once daily formulations containing cobicistat, stavudine, or on regimens containing stavudine or zidovudine* It is recommended to utilize a regimen of the integrase inhibitor, dolutegravir, combined with either disoproxil fumarate/emtricitabine or dolutegravir combined with tenofovir alafenamide/emtricitabine Cluster of differentiation (CD)4 count >= 200 within 120 days prior to enrollment or plasma HIV-1 RNA < 200 copies/mL within 120 days prior to randomization Patients with known human immunodeficiency virus (HIV) infection are eligible if not on antiviral agents and cluster of differentiation (CD)4 counts are adequate (>= 500) Patients with a known history of human immunodeficiency virus (HIV) seropositivity:* Must have undetectable viral load using standard HIV assays in clinical practice* Must have cluster of differentiation (CD)4 count >= 400/mcL* Must not require prophylaxis for any opportunistic infections (i.e., fungal, Mycobacterium avium complex [mAC], or pneumocystis jiroveci pneumonia [PCP] prophylaxis)* Must not be newly diagnosed within 12 months prior to sub-study registration Patients known to be human immunodeficiency virus (HIV) positive with one or more of the following:* Baseline cluster of differentiation (CD)4 count of < 250 cells/mm^3* History of acquired immune deficiency syndrome (AIDS) indicator conditions* Anti-retroviral therapy with any potent CYP3A4 inhibitor Known human immunodeficiency virus (HIV)-positive patients should have a cluster of differentiation (CD)4 count > 250/mm^3 Patients with known human immunodeficiency virus (HIV) may be eligible providing they meet the following additional criteria:* Cluster of differentiation (CD)4 cells >= 500/uL* Serum HIV viral load of < 25,000 IU/ml* No current antiretroviral therapy** Tests must be obtained within 28 days prior to registration; patients who are HIV positive (+) and do not meet all of these criteria are not eligible for this study (HIV/hepatitis testing are not required for patients without known infection) Patients who are human immunodeficiency virus (HIV) positive may participate if they meet the following eligibility requirements:* They must be stable on their anti-retroviral regimen, and they must be healthy from an HIV perspective* They must have a cluster of differentiation (CD)4 count of greater than 250 cells/mcL* They must not be receiving prophylactic therapy for an opportunistic infection Human immunodeficiency virus (HIV)-positive patients are eligible if on stable dose of highly active antiretroviral therapy (HAART), cluster of differentiation (CD)4 counts are greater than 350 and viral load is undetectable In order for patients with known history of testing positive for human immunodeficiency virus (HIV) or known acquired immunodeficiency syndrome (AIDS) to be eligible, they must be on a stable highly active antiretroviral therapy (HAART) regimen, have cluster of differentiation (CD)4 counts > 350, with no detectable viral load on quantitative polymerase chain reaction (PCR) Human immunodeficiency virus (HIV)-positive patients on combination antiretroviral therapy are eligible provided that they meet the following criteria in addition to the other protocol criteria: * Cancer as the only acquired immunodeficiency syndrome (AIDS)-defining condition* Cluster of differentiation (CD)4 cell count >= 250* Treatment sensitive HIV and prospects for long term survival on the basis of HIV disease alone* Willing to take anti-HIV therapy that will have minimal potential for pharmacokinetic interactions with GDC-0449 If HIV-positive, any cluster of differentiation (CD)4 count will be allowed on study Patients who are human immunodeficiency virus (HIV)-positive are eligible if: * CD4+ cell count greater or equal to 250 cells/mm^3* If patient is on antiretroviral therapy, there must be minimal interactions or overlapping toxicity of the antiretroviral therapy with the experimental cancer treatment; for experimental cancer therapeutics with CYP3A/4 interactions, protease inhibitor therapy is disallowed; suggested regimens to replace protease inhibitor therapy include dolutegravir given with tenofovir/emtricitabine; raltegravir given with tenofovir and emtricitabine; once daily combinations that use pharmacologic boosters may not be used* No history of non-malignancy acquired immune deficiency syndrome (AIDS)-defining conditions other than historical low CD4+ cell counts* Probable long-term survival with HIV if cancer were not present Human immunodeficiency virus (HIV)-positive patients on combination antiretroviral therapy are ineligible; however, HIV patients treated with regimens that have low cytochrome P450 (CYP450) inhibition may be allowed as long as the patients general health and cluster of differentiation (CD)4 counts are within acceptable levels Patients known to be human immunodeficiency virus (HIV) positive with a baseline cluster of differentiation (CD)4 count of < 250 cells/mm^3 or have a history of acquired immune deficiency syndrome (AIDS) indicator conditions Human immunodeficiency virus (HIV)-positive patients on antiretroviral medications that are CYP3A4 substrates will be closely monitored; HIV-positive patients will be excluded if they have a cluster of differentiation 4 (CD4) count < 200 Human immunodeficiency virus (HIV)-infected persons are eligible if they meet other eligibility criteria including the following:* No prior acquired immune deficiency syndrome (AIDS)-defining condition other than cluster of differentiation (CD)4+ cells nadir < 200/mm^3* Pre-leukemia CD4+ cell count >= 250/mm^3* Willing to adhere to antiretroviral therapy regimen with minimal overlapping toxicity and PK interactions with the experimental agents in this study; no zidovudine- and no ritonavir-containing regimens and no 3-drugs-in-1 pill regimens containing pharmacologic boosters are allowed; recommended regimens are integrase inhibitors combined with tenofovir and emtricitabine Human immunodeficiency virus (HIV) positive (+) patients with cluster of differentiation 4 (CD4) counts >= 250 cells/mm^3 on anti-viral therapy Human immunodeficiency (HIV) positive (+) patients are eligible provided they meet the other eligibility criteria and:* Cluster of differentiation (CD)4+ cells are >= 250/mm^3* There is no history of acquired immunodeficiency syndrome (AIDS) defining conditions other than historically low CD4+ cell count* The following antiretroviral agents are not allowed: zidovudine, stavudine, protease inhibitors, non-nucleoside reverse transcriptase inhibitors, combination pills with pharmacologic boosters* Recommended antiretroviral regimens to avoid pharmacokinetic (PK) interactions include strand integrase inhibitors with nucleoside reverse transcriptase inhibitors (for example, dolutegravir given with tenofovir and emtricitabine) Human immunodeficiency virus (HIV)-positive patients on combination antiretroviral therapy are eligible; unless the patients cluster of differentiation (CD)4+ count is below the institutional lower limit of normal Patients with a known history of human immunodeficiency virus (HIV) are eligible, if they meet all of the following conditions:* No history of HIV complications with the exception of cluster of differentiation (CD)4 count < 200 cells/mm^3* No antiretroviral therapy with overlapping toxicity such as myelosuppression* HIV viral loads below the limit of detection* No history of highly active antiretroviral therapy (HAART)-resistant HIV Patients with any stage of pathologically confirmed cluster of differentiation (CD)3+ acute, lymphoma, chronic, or smoldering subtypes of ATLL Patients with acquired immunodeficiency syndrome (AIDS-related illnesses) or known human immunodeficiency virus (HIV) disease must:* Have a cluster of differentiation (CD)4 count >= 200 cells/uL within 30 days before beginning study therapy* Be off all antiretroviral therapy (prophylaxis/treatment) more than 60 days before beginning study therapy, and* Have no evidence of opportunistic infections Patients who are known to be human immunodeficiency virus (HIV)-positive at registration are eligible at the time of registration:1. Cluster of differentiation (CD)4+ cell count greater or equal to 250 cells/mm^32. If patient is on antiretroviral therapy, there must be minimal interactions or overlapping toxicity of the antiretroviral therapy with the experimental cancer treatment; once daily combinations that use pharmacologic boosters may not be used3. No history of non-malignancy acquired immunodeficiency syndrome (AIDS)-defining conditions other than historical low CD4+ cell counts Patients with hepatitis B virus infection must have undetectable hepatitis B virus (HBV) on suppressive therapy and no evidence of HBV-related hepatic damage; patients with hepatitis C virus infection are eligible if complete eradication therapy has been successfully completed, and there is no detectable hepatitis C virus (HVC) or related hepatic damage; patients with known human immunodeficiency virus (HIV) infection are eligible if they meet all of the following criteria:* Patient must have no history of acquired immune deficiency syndrome (AIDS)-related complications, other than a history of low CD4+ T-cell count (< 200/mm3) prior to initiation of combination antiretroviral therapy; on study CD4+ T-cell count may not be informative due to leukemia and should not be used as an exclusion criterion if low* Patient must be healthy on the basis of HIV disease with high likelihood of near normal life span were it not for the leukemia* Patient must have serum HIV viral load of < 200 copies/mm^3* Patient must be on combination antiretroviral therapy with minimal pharmacokinetic interactions with study therapy and minimal overlapping clinical toxicity with protocol therapy; (recommend a regimen of the integrase inhibitor dolutegravir combined with either disoproxil fumarate/emtricitabine or dolutegravir combined with tenofovir alafenamide/emtricitabine)* Protease inhibitors and once daily formulations containing cobicistat are NOT allowed* Stavudine and zidovudine (AZT) are NOT allowed