The goal of this clinical research study is to learn if it is safe and feasible to transplant changed cord blood for patients with leukemia or lymphoma. Researchers also want to learn if this can help to control the disease.

The cord blood will be changed to make use of sugar that is found in small amounts in blood cells. It plays a role in signaling where in the body the transplanted cells should go to. Adding more sugars to the cord blood cells in the laboratory is designed to help the cord blood cells find their way faster to the bone marrow. This may help your blood counts to recover faster. This process is called fucosylation.

Anti-thymocyte globulin (ATG) is a protein that removes immune cells that cause damage to the body.

Clofarabine is designed to interfere with the growth and development of cancer cells.

Fludarabine is designed to interfere with the DNA (genetic material) of cancer cells, which may cause the cancer cells to die. This chemotherapy is also designed to block your body's ability to reject the donor's bone marrow cells.

Melphalan and busulfan are designed to bind to the DNA of cells, which may cause cancer cells to die.

Mycophenolate mofetil (MMF) and tacrolimus are designed to block the donor cells from growing and spreading in a way that could cause graft versus host disease (GVHD -- a condition in which transplanted tissue attacks the recipient's body). This may help to prevent GVHD.

Rituximab is designed to attach to cancer cells, which may cause them to die.

Central Venous Catheter Placement:

You will first have a central venous catheter (CVC) placed. A CVC is a sterile flexible tube that will be placed into a large vein while you are under local anesthesia. Your doctor will explain this procedure to you in more detail, and you will be required to sign a separate consent form for it.

The chemotherapy, some of the other drugs in this study, and the cord blood transplant will be given by vein through your CVC. Blood samples will also be drawn through your CVC. The CVC will remain in your body for about 2-5 months.

Study Plans:

If you agree to take part in this study, your doctor will choose one of the following 2 study plans based on the disease and your age and medical history.

Fludarabine/Clofarabine/Busulfan/Rituximab/Total Body Irradiation:

If you are between 1 and 55 years of age and can receive high-dose chemotherapy, or you are between 55 and 65 years old and your doctor agrees, you will receive fludarabine, clofarabine, busulfan, ATG, total body irradiation, and possibly rituximab.

You will receive a test dose of busulfan by vein over 60 minutes as an outpatient. If the test dose cannot be given as an outpatient, you will be admitted to the hospital on Day -10 for intravenous (IV) fluids and will receive the busulfan test dose on Day -9. This low-level "test" dose of busulfan is to check how the level of busulfan in your blood changes over time. This information will be used to decide the next dose needed to reach the target blood level that matches your body size.

About 11 samples of blood will be drawn for pharmacokinetic (PK) testing of busulfan. PK testing measures the amount of study drug in the body at different time points. These blood samples will be drawn at various timepoints starting before the busulfan infusion and continuing over about the next 11 hours. The blood samples will be repeated again with the first day of high-dose busulfan treatment. Each sample will be about 1 teaspoon of blood. A temporary heparin lock will be placed in your vein to lower the number of needle sticks needed for these draws. If it is not possible for the PK tests to be performed for technical or scheduling reasons, you will receive the standard fixed dose of busulfan.

If you will receive the test dose as an outpatient, you will be admitted to the hospital on Day -8 and will receive fluids by vein. If appropriate for the disease, you will receive rituximab by vein over 4-6 hours on Day -8.

If you will receive the test dose as an inpatient, you will be admitted on Day -10 and will receive fluids by vein. If appropriate for the disease, you will receive rituximab by vein over 4-6 hours on Day -10.

On Days -7 through -4, you will receive fludarabine by vein over 1 hour, clofarabine by vein over 1 hour, and busulfan by vein over 3 hours. You will receive ATG on Days -4 and -3. On Day -3, you will receive a single treatment of low-dose total body irradiation. You will "rest" (not receive chemotherapy drugs) on Days -2 and -1. Day 0 is the day of the cord blood transplant, so the negative day numbers are used to label the treatment days before the transplant.

All chemotherapy drugs, fluids, and other drugs that must be given by vein will be infused through the catheter. Once the back-up cells are collected, all participants will be admitted to the hospital as indicated by their assigned treatment plan schedule.

Fludarabine/Melphalan:

If your doctor chooses fludarabine and melphalan, you will have the following schedule:

• On Day -6 (6 days before your transplant), you will be admitted to the hospital and will receive fluids by vein.
• On Days -5, -4, and -3, you will receive fludarabine by vein over 30 minutes.
• On Days -4 and -3, you will receive ATG by vein over 4 hours.
• On Day -2, you will receive fludarabine by vein over 30 minutes and melphalan by vein over 30 minutes.
• On Day -1, you will not receive any drugs.
• On Day 0, you will receive your cord blood transplant through the CVC.

Supportive Drugs:

Starting on Day -3, you will receive mycophenolate mofetil as a tablet by mouth 2 times a day. If you are not able to take the tablet by mouth, you will receive MMF by vein over 2 hours 2 times a day. If you do not have GVHD at Day 100 after your cord blood transplant, the dose of MMF will be gradually lowered. If you have GVHD, MMF may be stopped 7 days after the GVHD is controlled.

Starting on Day -2, you will receive tacrolimus by vein as a continuous (nonstop) infusion until you are able to take it by mouth. You will then take tacrolimus by mouth 2 times a day for about 6 months. After that, your tacrolimus dose may be gradually lowered if you do not have GVHD. Your doctor will discuss this with you.

Starting on Day 0, you will receive filgrastim (G-CSF) through a needle under the skin 1 time a day every day until your white blood count begins to recover. Filgrastim is designed to help cells in the bone marrow to divide, which helps raise white blood cells counts more quickly, lower fever, and decrease the risk of infection.

Study Visits:

At about 1, 3, 6, and 12 months after the transplant:

• You will have a physical exam.
• You will be checked for possible reactions to the transplant and study drugs, including GVHD.
• Blood (about 4 teaspoons) will be drawn for routine tests, to check for CMV antibodies, and for genetic tests to learn if the donor's cells have "taken". The routine blood tests will be repeated as often as the doctor thinks is needed.
• If the doctor thinks it is needed, you will have a bone marrow aspiration to check the status of the disease.

You will need to stay in the hospital for about 4 weeks. After you leave the hospital, you will continue as an outpatient in the hospital area, which means you will have to stay close enough to be able to come back for any visits for about 100 days after the transplant.

Length of Participation:

You will be on study for up to 1 year. You will be taken off study early if the disease gets worse, if intolerable side effects occur, if the cord blood is infected and cannot be transplanted, if you are unable to follow study directions, or if your doctor thinks it is in your best interest.

This is an investigational study. Fucosylation is not an FDA-approved process. It is currently being used for research purposes only. Fludarabine, busulfan, clofarabine, melphalan, mycophenolate mofetil, tacrolimus, and rituximab are FDA approved and commercially available to be given to patients with leukemia or lymphoma having a cord blood transplant. Total body irradiation is delivered using FDA-approved and commercially available methods.

Up to 50 patients will be enrolled in this study. All will be enrolled at MD Anderson.

Inclusion Criteria:

1. Patients must have one of the following hematologic malignancies: Acute Myelogenous Leukemia (AML), induction failure, high-risk for relapse first remission (with intermediate-risk or high-risk cytogenetics, flt3 mutation positive and/or evidence of minimal residual disease by flow cytometry), secondary leukemia from prior chemotherapy and/or arising from MDS, Langerhan's cell histiocytosis, any disease beyond first remission; or,
2. Myelodysplastic Syndrome (MDS): Primary or therapy related; or,
3. Acute Lymphoblastic Leukemia (ALL): induction failure, primary refractory to treatment (do not achieve complete remission after first course of therapy) or are beyond first remission including second or greater remission or active disease. Patients in first remission are eligible if they are considered high risk, defined as any of the following detected at any time: with translocations 9;22 or 4;11, hypodiploidy, complex karyotype, secondary leukemia developing after cytotoxic drug exposure, and/or evidence of minimal residual disease, or acute biphenotypic leukemia, or double hit non-Hodgkin's lymphoma; or,
4. Non-Hodgkin's Lymphoma (NHL) in primary induction failure, second or third complete remission, refractory disease, or relapse (including relapse post autologous hematopoietic stem cell transplant). Double hit lymphomas in first remission or more advanced disease; or,
5. Small Lymphocytic Lymphoma (SLL), or Chronic Lymphocytic Leukemia (CLL) with progressive disease following standard therapy; or,
6. CML second chronic phase or accelerated phase; or,
7. Hodgkin's Disease (HD): Induction failures, second or third complete remission, or relapse (including relapse post autologous hematopoietic stem cell transplant).
8. Patients Age Criteria: Age >/= 1 and </= 80 years old. Eligibility for pediatric patients will be determined in conjunction with an MDACC pediatrician.
9. Performance score of at least 80% by Karnofsky or PS < 3 (ECOG) (age >/= 12 years), or Lansky Play-Performance Scale of at least 60% or greater (age <12 years).
10. Adequate major organ system function as demonstrated by: a. Left ventricular ejection fraction of at least 40-45% b. Pulmonary function test (PFT) demonstrating a diffusion capacity of least 50% predicted. For children </= 7 years of age who are unable to perform PFT, oxygen saturation >/= 92% on room air by pulse oximetry. c. Creatinine < 1.6 mg/dL. d. SGPT/bilirubin \