The goal of this clinical research study is to find the highest tolerable dose of Avastin™ (bevacizumab) and Velcade™ (bortezomib) that can be given in combination to patients with a metastatic or unresectable advanced malignancy. The safety and effectiveness of this drug combination will also be studied.

Bevacizumab is an anti-cancer drug designed to prevent or slow down the growth of cancer cells by blocking blood vessels that supply nutrients necessary for tumor growth.

Bortezomib is an anti-cancer drug designed to block the proteins needed for tumor growth. This may cause cancer cells to die.

If you are found to be eligible to take part in this study, you will be enrolled into a group of about 6 participants. The first group of participants will receive the lowest dose of bevacizumab and bortezomib combined. The next group of participants will receive the next highest dose of bevacizumab and bortezomib combined. This process will continue until the study doctor finds the highest dose that can be tolerated. The dose that you receive will depend on when you are enrolled in this study and the safety data that is available at that time. The dose of bevacizumab and bortezomib that you receive may be lowered if you do not tolerate the study drug combination well. You will not receive any doses of the study drug higher than the dose you are first assigned to.

Bevacizumab and bortezomib will be given in "cycles." Cycles will be about 21 days long or longer, depending on any side effects you may experience. During Cycle 1, Day 1, you will receive bevacizumab by vein over 90 minutes. If bevacizumab is well tolerated in Cycle 1, it will given over 60 minutes in Cycle 2. If it is well tolerated in Cycle 2, it will be given over 30 minutes in Cycle 3. It will continue to be given over 30 minutes in further cycles as long as the drug is still being well tolerated. Depending on which dose level you are assigned to, you will receive bortezomib on Days 1 and 8, or on Days 1, 4, 8, and 11. You will receive bortezomib by vein over about 1-5 minutes.

You will have blood drawn (about 1 tablespoon each time) for routine tests once a week during Cycle 1. You will have a physical exam sometime between Days 7 to 14 during Cycle 1. During the rest of the cycles, you will have a physical exam and blood drawn (about 1 tablespoon each time) for routine tests once every 3 weeks. The status of the disease will be measured by a CT or MRI scan after every 2 cycles.

Once the highest tolerable dose (maximum tolerated dose or MTD) of the combination of bevacizumab and bortezomib is found, up to 15 additional participants with advanced cancer will be enrolled to receive that dose, so that researchers can learn more about the effects of the study drugs on the tumor. A tumor biopsy will be required within two weeks before the first treatment and again at the end of the first cycle for those patients. Up to 10 additional patients with kidney cancer will be enrolled to receive the highest tolerable dose. Tumor biopsies are optional for patients with kidney cancer.

You may continue to receive bevacizumab and bortezomib on this study, unless the cancer gets worse or you experience any intolerable side effects.

Once your participation is over in this study, you will receive standard of care follow-up for the disease.

This is an investigational study. Bevacizumab and bortezomib are both FDA approved and commercially available. Bevacizumab is FDA approved for the treatment of colorectal cancer. Bortezomib is FDA approved for the treatment of multiple myeloma. The combination use of bevacizumab and bortezomib is investigational and authorized for use in research only.

Up to 111 patients will take part in this study. All will be enrolled at MD Anderson.

Inclusion Criteria:

1. Patients with advanced or metastatic cancer that is refractory to standard therapy, relapsed after standard therapy, or have no standard therapy that induces a complete response (CR) rate of at least 10% or improves survival by at least three months.
2. Patients must be >/= 6 weeks beyond treatment with nitrosoureas or mitomycin-C, >/= 4 weeks beyond other chemo- or radiotherapy, and must have recovered to </= grade 1 toxicity for any treatment-limiting toxicity of prior therapy. (Exception: patients who received palliative low dose radiotherapy to the limbs 1-4 weeks before this therapy provided pelvis, ribs, sternum, scapulae, vertebrae or skull were not included in the radiotherapy field). Patients who have received non-chemotherapeutic biologic agents must wait 5 half-lives or 4 weeks, whichever is shorter, from the last day of treatment.
3. The Eastern Cooperative Oncology Group (ECOG) performance status </= 2 (Karnofsky >/= 60%).
4. Patients must have normal organ and marrow function defined as: leukocytes >/= 3,000/mL; absolute neutrophil count >/= 1,500/mL; platelets >/=75,000/mL; creatinine \