This study will examine whether the medication abatacept is safe in the treatment of giant cell arteritis (GCA) and Takayasu's arteritis (TAK) and whether there is evidence of effectiveness to support pursuing this medication in larger scale studies. During the course of this study, blood samples will be obtained for research purposes to examine the mechanisms that occur in GCA and TAK and the effects of atabacept. GAC and TAK are diseases that share a number of common features and are both characterized by blood vessel inflammation (vasculitis) that can be serious or even life threatening. While glucocorticoid medications, such as prednisone, can improve the symptoms of GCA and TAK, these are associated with significant side effects and they do not prevent disease recurrences in most people. Abatacept is a medication that is currently approved for the treatment of rheumatoid arthritis. Abatacept is believed to work by interfering with the action of inflammatory cells called activated T cells. Evidence form laboratory research has supported that activated T cells may play a role in the blood vessel inflammation seen in GCA and TAK. Because of this information and the need to identify better treatment for GCA and TAK, conducting a study to explore the safety and possible effectiveness of abatacept is an appropriate next step.