Function theory on Euclidean domains in relation to potential theory, partial differential equations, probability, and harmonic analysis has been the target of investigation for decades. There is a wealth of classical literature in the subject. Geometers began to study function theory with the primary reason to prove a uniformization type theorem in higher dimensions. It was first proposed by GreeneWu and Yau to study the existence of bounded harmonic functions on a complete manifold with negative curvature. While uniformization in dimension greater than 2 still remains an open problem, the subject of function theory on complete manifolds takes on life of its own. The seminal work of Yau [Y1] provided a fundamental technique in handling analysis on noncompact, complete manifolds. It also opens up many interesting problems which are essential for the understanding of analysis on complete manifolds. Since Yau’s paper in 1975, there are many developments in this subject. The aim of this article is to give a rough outline of the history of a specific point of view in this area, namely, the interplay between the geometry – primarily the curvature – and the function theory. Throughout this article, unless otherwise stated, we will assume that M is an n-dimensional, complete, noncompact, Riemannian manifold without boundary. In this case, we will simply say that M is a complete manifold. One of the goal of this survey is to demonstrate, by way of known theorems, the two major steps which are common in many geometric analysis programs. First, we will show how one can use assumptions on the curvature to conclude function theoretic properties of the manifold M. Secondly, we will showed that function theoretic properties can in turn be used to conclude geometrical and topological statements about the manifold. In many incidents, combining the two steps will result in a theorem which hypothesizes on the curvature and concludes on either the topological, geometrical, or complex structure of the manifold. The references will not be comprehensive due to the vast literature in the subject. It is merely an indication of the flavor of the field for the purpose of whetting one’s appetite. As examples of areas not being discussed in this note are harmonic analysis (function theory) on symmetric spaces, Lie groups, and discrete groups. The contributors to this subject are Furstenberg, Varopoulos, Coulhon, Saloff-Coste, and