GCA and TAK both cause inflammation in the lining of the arteries, which can interfere with the body's ability to carry oxygen to areas that need it. Symptoms of GCA include headaches, jaw pain, and blurred or double vision. Serious symptoms that occur less commonly are blindness and stroke. TAK symptoms include fever, fatigue, weight loss, arthritis, and non-specific aches and pains. There may also be tenderness near affected arteries. Researchers believe that GCA and TAK are diseases that are controlled by the body's immune system. Activated T-cells, specifically, are critical to the origin and development of these diseases. Abatacept is a medication that modulates the signal required for T-cell activation. This study will evaluate the safety and effectiveness of abatacept in treating GCA and TAK and preventing disease relapse.