Abstract: The Aleksandrov-Bakelman-Pucci Maximum Principle is a pointwise estimate in elliptic PDE theory. My main result is achieving nonlocal version of ABP estimate on some manifolds. In this talk, I will briefly review classical PDE regularity theory on both divergent and non-divergent form. After that, I will introduce nonlocal elliptic ABP estimate on Euclidean space and on some manifolds.