We will present recent developments for lattice regularization of quantum field theories on curved Riemann manifolds. A classical lattice action for scalar fields which incorporates elements of Regge calculus, finite elements, and discrete exterior calculus will be discussed in an organized framework applicable to any number of dimensions, and its convergence properties will be studied. We will demonstrate how the Wilson-Dirac operator may be constructed on a two dimensional curved lattice and study the free Dirac theory on the 2-sphere.
A renormalization scheme will be presented in which quantum loops sensitive to the underlying curvature are canceled by explicit counterterms allowing one to reach the continuum limit in interacting theories. A Monte Carlo study of phi^4 theory on the 2-sphere will be presented as a demonstration of the validity of the formalism. Ongoing work may be mentioned, including a study of the 3D Ising fixed point in radial quantization. We will remark on prospects for applications to conformal gauge theories in four dimensions.