The aim of the research in this study is to make participants' transplant safer by reducing the risk of developing GVHD and GVHD-related complications by giving participants a dose of the drug tocilizumab in addition to the standard approach for GVHD prevention. Tocilizumab reduces the risk of inflammation by blocking the effect of Interleukin-6, a protein that exists in high levels in the blood when there is inflammation. Participants who receive stem cell transplants have high levels of this protein in their blood early after transplant. Therefore, the goal of this study is to reduce the risk of inflammation after transplant with the addition of Tocilizumab. This could decrease the risk of developing GVHD and GVHD-associated complications.

Inclusion Criteria:

I. Acute myelogenous leukemia (AML)

• Complete first remission (CR1) at high risk for relapse such as any of the following:
• Known prior diagnosis of myelodysplasia (MDS) or myeloproliferative disorder
• Therapy-related AML
• White cell count at presentation > 100,000
• Presence of extramedullary leukemia at diagnosis
• Any unfavorable subtype by FAB or WHO classification
• High-risk cytogenetics (e.g. those associated with MDS, abnormalities of 5, 7, 8, complex karyotype) or high risk molecular abnormalities
• Requirement for 2 or more induction to achieve CR1
• Any patient with newly diagnosed AML with intermediate risk cytogenetics who elects allograft with curative intent over consolidation chemotherapy
• Any patient unable to tolerate consolidation chemotherapy as would have been deemed appropriate by the treating physician
• Other high risk features not defined above
• Complete second remission (CR2)
• Primary refractory or relapsed AML with less than 10% blasts before transplant. Persistent/relapsed AML with cytogenetic, flow cytometric, or molecular aberrations in >/= 10% of cells are eligible

II. Acute lymphoblastic leukemia (ALL)

• Complete first remission (CR1) at high risk for relapse such as any of the following:
• White cell count at presentation > 30,000 for B-cell lineage and > 100,00 for T-cell lineage
• Presence of any high-risk cytogenetic abnormalities such as t(9;22), t(1;19), t(4;11) or other MLL rearrangements (11q23) or other high-risk molecular abnormality
• Failure to achieve complete remission after four weeks of induction therapy
• Persistence or recurrence of minimal residual disease on therapy
• Any patient with newly diagnosed ALL >/= 50 years-old
• Any patient unable to tolerate consolidation and/or maintenance chemotherapy as would have been deemed appropriate by the treating physician
• Other high risk features not defined above
• Complete second remission (CR2)
• Primary refractory or relapsed ALL with less than 5% blasts before transplant. Persistent/relapsed ALL with cytogenetic, flow cytometric or molecular aberrations in >/=5% of cells are eligible.

III. Other acute leukemias: leukemias of ambiguous lineage or of other types e.g. blastic plasmacytoid dendritic cell neoplasm with less than 5% blasts. Persistent/relapsed disease with cytogenetic, flow cytometric or molecular aberrations in >/=5% of cells are eligible

IV. Myelodysplastic Syndrome (MDS)/ Myeloproliferative Disorders (MPD) other than myelofibrosis:

• International prognostic scoring system (IPSS) risk score of INT-2 or high risk at the time of diagnosis
• Any IPSS risk category if life-threatening cytopenia(s) exists
• Any IPSS risk category with karyotype or genomic changes that indicate high risk for progression to acute myelogenous leukemia
• MDS/ myeloproliferative disorder overlap syndromes without myelofibrosis
• MDS/ MPD patients must have less than 10% bone marrow myeloblasts and ANC >/= 0.2 (growth factor supported if necessary) at transplant work-up

V. Non-Hodgkin lymphoma (NHL) or Hodgkin lymphoma (HL) at high-risk of relapse or progression if not in remission:

• Eligible patients with aggressive histologies (such as, but not limited to, diffuse large B-cell NHL, mantle cell NHL, and T-cell histologies) in CR
• Eligible patients with indolent B cell NHL (such as, but not limited to, follicular, small cell or marginal zone NHL) will have 2nd or subsequent progression with stable disease/CR/PR with no single lesion equal to or more than 5 cm.
• Eligible patients with HL will be without progression of disease (POD) after salvage chemotherapy with no single lesion equal to or more than 5cm.

Organ Function and Performance Status Criteria:

• Karnofsky score >/= 70% (inpatient Leukemia service transfers without discharge are acceptable provided patient has equivalent KPS as if were outpatient)
• Calculated creatinine clearance >/= 60 ml/min
• Bilirubin < 1.5 mg/dL (unless benign congenital hyperbilirubinemia)
• ALT </= 3 x upper limit of normal
• Pulmonary function (spirometry and corrected DLCO) >/= 50% predicted
• Left ventricular ejection fraction greater than 50%
• Albumin >/= 3.0
• Age-adjusted Hematopoietic Cell Transplantation-Comorbidity Index (aaHCT-CI) less than or equal to 7

Graft Criteria:

2 CB units will be selected according to current MSKCC unit selection algorithm. High resolution 8 allele HLA typing and recipient HLA antibody profile will be performed. Unit selection will occur based on HLA-match, total nucleated cell (TNC) and CD34+ cell dose adjusted per patient body weight. The bank of origin will also be taken into account. Donor specific HLA antibodies, if present, will also be taken into consideration and may influence the selection of the graft.

• Each CB unit must be at least 3/8 HLA-matched to the patient considering high resolution 8-allele HLA typing
• Each CB unit will be required to have a cryopreserved TNC dose of at least 1.5 x 10^7 TNC/ recipient body weight (TNC/ kg)
• Each CB unit will be required to have a cryopreserved CD34+ cell dose of at least 1.0 x 10^5 CD34+ cells/ recipient body weight (CD34+/kg).
• A minimum of one domestic will be reserved as a backup unit.

Exclusion Criteria:

• Indolent NHL or Hodgkin lymphoma with POD after most recent salvage chemotherapy
• Diagnosis of myelofibrosis or other malignancy with moderate-severe bone marrow fibrosis
• Any diagnosis without prior immunosuppressive chemotherapy within 3 months of intended admission for transplant
• Prior checkpoint inhibitors/ blockade in the last 12 months
• Two prior stem cell transplants of any kind
• One prior autologous stem cell transplant within the preceding 12 months
• One prior allogeneic stem cell transplant within the preceding 24 months
• Prior radiation therapy with 400cGy or more of TBI
• Active and uncontrolled infection at time of transplantation
• HIV infection
• Seropositivity for HTLV-1.
• Inadequate performance status/ organ function.
• Pregnancy or breast feeding
• Patient or guardian unable to give informed consent or unable to comply with the treatment protocol including appropriate supportive care, long-term follow-up, and research tests.