Safety and Efficacy Study of the Medtronic CoreValve® System in the Treatment of Symptomatic Severe Aortic Stenosis in High Risk and Very High Risk Subjects Who Need Aortic Valve Replacement

Inclusion Criteria: * Extreme Risk Only: Subject must have comorbidities such that one cardiologist and two cardiac surgeons agree that medical factors preclude operation, based on a conclusion that the probability of death or serious morbidity exceeds the probability of meaningful improvement. Specifically, the predicted operative risk of death or serious, irreversible morbidity is ≥ 50% at 30 days. * High Risk Surgical Only: Subject must have comorbidities such that one cardiologist and two cardiac surgeons agree that predicted risk of operative mortality is ≥15% (and predicted operative mortality or serious, irreversible morbidity risk of \< 50%) at 30 days. * Subject has senile degenerative aortic valve stenosis with: mean gradient \> 40 mmHg, or jet velocity greater than 4.0 m/sec by either resting or dobutamine stress echocardiogram, or simultaneous pressure recordings at cardiac catheterization (either resting or dobutamine stress), AND an initial aortic valve area of ≤ 0.8 cm2 (or aortic valve area index ≤ 0.5 cm2/m2) by resting echocardiogram or simultaneous pressure recordings at cardiac catheterization * Subject is symptomatic from his/her aortic valve stenosis, as demonstrated by New York Heart Association (NYHA) Functional Class II or greater. * The subject or the subject's legal representative has been informed of the nature of the trial, agrees to its provisions and has provided written informed consent as approved by the IRB of the respective clinical site. * The subject and the treating physician agree that the subject will return for all required post-procedure follow-up visits. Exclusion Criteria: Clinical * Evidence of an acute myocardial infarction ≤ 30 days before the intended treatment. * Any percutaneous coronary or peripheral interventional procedure performed within 30 days prior to the intended treatment. (High Risk Surgical Only: Including bare metal stents. Additionally, any drug eluting stents placed within 6 months prior to the index procedure.) * Blood dyscrasias as defined: leukopenia (WBC \< 1000mm3), thrombocytopenia (platelet count \<50,000 cells/mm3), history of bleeding diathesis or coagulopathy. * Untreated clinically significant coronary artery disease requiring revascularization. * Cardiogenic shock manifested by low cardiac output, vasopressor dependence, or mechanical hemodynamic support. * Need for emergency surgery for any reason. * Severe ventricular dysfunction with left ventricular ejection fraction (LVEF) \< 20% as measured by resting echocardiogram. * Recent (within 6 months) cerebrovascular accident (CVA) or transient ischemic attack (TIA). * End stage renal disease requiring chronic dialysis or creatinine clearance \< 20 cc/min. * Active Gastrointestinal (GI) bleeding within the past 3 months. * A known hypersensitivity or contraindication to any of the following which cannot be adequately pre-medicated: * Aspirin * Heparin (HIT/HITTS) * Bivalirudin (Extreme Risk Only) * Nitinol (titanium or nickel) * Ticlopidine and clopidogrel * Contrast media * Ongoing sepsis, including active endocarditis. * Subject refuses a blood transfusion. * Life expectancy \< 12 months due to associated non-cardiac co-morbid conditions. * Other medical, social, or psychological conditions that in the opinion of an Investigator precludes the subject from appropriate consent. * Severe dementia (resulting in either inability to provide informed consent for the trial/procedure, prevents independent lifestyle outside of a chronic care facility, or will fundamentally complicate rehabilitation from the procedure or compliance with follow-up visits). * Currently participating in an investigational drug or another device trial. * Symptomatic carotid or vertebral artery disease. * Subject has been offered surgical aortic valve replacement but declined. (High Risk Surgical Only) Anatomical * Native aortic annulus size \< 18 mm or \> 29 mm per the baseline diagnostic imaging. * Pre-existing prosthetic heart valve any position. * Mixed aortic valve disease (aortic stenosis and aortic regurgitation with predominant aortic regurgitation (3-4+)). * Moderate to severe (3-4+) or severe (4+) mitral or severe (4+) tricuspid regurgitation. * Moderate to severe mitral stenosis. * Hypertrophic obstructive cardiomyopathy. * New or untreated echocardiographic evidence of intracardiac mass, thrombus or vegetation. * Severe basal septal hypertrophy with an outflow gradient. * Aortic root angulation (angle between plane of aortic valve annulus and horizontal plane/vertebrae) \> 70° (for femoral and left subclavian/axillary access) and \> 30° (for right subclavian/axillary access). * Ascending aorta diameter \> 43 mm if the aortic annulus diameter is 23-29 mm; ascending aorta diameter \> 40 mm if the aortic annulus diameter is 20-23 mm; or an ascending aorta diameter \> 34 mm if the aortic annulus diameter is 18-20 mm. * Congenital bicuspid or unicuspid valve verified by echocardiography. * Sinus of valsalva anatomy that would prevent adequate coronary perfusion. Vascular \- Transarterial access not able to accommodate an 18Fr sheath.