Abdominal aortic aneurysm (AAA) is a condition whereby the terminal aorta permanently dilates to dangerous proportions, risking rupture. Rupture of AAA is a significant cause of death in this country that is generally avoided by surgical repair. In order to improve our understanding and diagnosis of this disease, we call upon first principles of finite elasticity. We use such principles to estimate the AAA wall strength and acting wall stress, which in turn allow evaluation of an individual aneurysm’s risk of rupture. In this paper, we summarize the work that has been performed to date in applying finite elasticity to abdominal aortic aneurysm research.