Local renormalization group (RG) is a generalization of standard RG techniques in quantum field theory such that coupling constants are promoted to spacetime-dependent functions. After giving an overview of some basics of local RG, we explain that local RG equations are manifested naturally in the context of the AdS/CFT correspondence. A key ingredient there is a Hamilton-Jacobi (HJ) formulation of classical gravity in AdS. We talk briefly about our recent work that is aimed at incorporating gauge fields into the HJ formalism in AdS gravity.