The goal of this clinical research study is to learn if the combination of fludarabine, cyclophosphamide, rituximab, and bevacizumab is effective in treating chronic lymphocytic leukemia in patients who have already been treated with chemotherapy. The safety of this treatment will also be studied.

During the study, you will have up to 6 "cycles" of treatment. A cycle is made up of treatment with the study drugs for 3-4 days and then about 3-1/2 weeks (25 days) of no treatment (about 4 weeks total). Treatment with the study drugs will be given for 4 days in a row on the first cycle, and 3 days in a row on Cycles 2-6. On Day 1 of each cycle, you will receive rituximab through a needle in a vein. On Cycle 1, since it is your first exposure to rituximab, it must be given slowly, so it may take 6-8 hours to complete. On the cycles after that, it can be given more rapidly, over 3-4 hours. Cyclophosphamide and fludarabine will be given separately through a needle in a vein on Days 2-4 of Cycle 1 and Days 1-3 of Cycles 2-6. Cyclophosphamide and fludarabine will each be given over 30 minutes. Bevacizumab will be given through a needle in a vein over 90 minutes on Day 3 of Cycle 1. If it is well tolerated, the next dose of Bevacizumab will be given over 60 minutes on Day 2 of Cycle 2. If the Cycle 2 dose is well tolerated, the next doses of Bevacizumab will be given over 30 minutes on Day 2 of Cycles 2-6. In addition to the study drugs, you may also be given fluids by vein to help flush the kidneys, to help prevent possible kidney damage. You may receive up to 6 cycles of treatment. Treatment will be given on an outpatient basis. The injections for each daily treatment visit should take less than 6 hours.

Up to 66 patients will take part in the study. All will be enrolled at M.D. Anderson.

Inclusion Criteria:

• Diagnosis of B-cell CLL
• Relapsed, fludarabine-sensitive (duration of response > 6 months as assessed by prior treating physician) or fludarabine-naive patients
• Rai Stage III or IV, or Rai Stage I or II if determined to have disease progression as evidenced by rapid doubling of peripheral lymphocyte count, progressive lymphadenopathy, progressive splenomegaly, or B symptoms.
• Prestudy WHO Performance Status </= 2.
• Signed, written Institutional Review Board (IRB)approved informed consent.
• Men and women of reproductive potential must agree to follow accepted birth control methods during treatment and for 3 months after completion of treatment.
• Acceptable liver function: Bilirubin </= 2.0 mg/dL (26 umol/L), AST (SGOT) and/or ALT (SGPT) </= 2 times upper limit of normal.
• Acceptable hematologic status: Platelet count >/= 50 x 10^9/L., absolute neutrophil count (ANC) >/= 1 x 10^9/L.
• Acceptable renal function: Serum creatinine </= 2.0 mg/dL

Exclusion Criteria:

• Cancer radiotherapy, radioimmunotherapy, biological therapy, chemotherapy, or other investigational therapy within 4 weeks prior to Study Day 1.
• Known infection with HIV, hepatitis B, or hepatitis C
• Transformation to aggressive B-cell malignancy (e.g., large B-cell lymphoma, Richter's Syndrome, or prolymphocyte leukemia [PLL]).
• Patients with secondary malignancy requiring active treatment (except hormonal therapy).
• Active uncontrolled bacterial, viral, or fungal infections.
• New York Heart Association Class II-IV cardiac disease or myocardial infarction within the past 6 months prior to Study Day 1.
• Pregnant or currently breast-feeding.
• Current, recent (within 4 weeks of the first infusion of this study), or planned participation in an experimental drug study other than a Genentech-sponsored bevacizumab cancer study.
• Blood pressure of > 150/100 mmHg
• Unstable angina
• History of stroke within 6 months
• Clinically significant peripheral vascular disease
• Evidence of bleeding diathesis or coagulopathy
• Major surgical procedure, open biopsy, or significant traumatic injury within 28 days prior to Day 1, anticipation of need for major surgical procedure during the course of the study.
• Minor surgical procedures, fine needle aspirations or core biopsies within 7 days prior to Day 1.
• Urine protein:creatinine ratio >/= 1.0 at screening.
• History of abdominal fistula, gastrointestinal perforation, or intra-abdominal abscess within 6 months prior to Day 1.
• Serious, non-healing wound, ulcer, or bone fracture.

Two participants included in the study demographics were registered but withdrew prior to receiving treatment.

Participants evaluated for adverse events at each study visit for study duration, up to six courses of 4 weeks. Overall collection period: May 2011 to September 2014.