In this talk, we will examine a family of nonlinear elliptic partial differential equations and closely related integral equations. The equations we consider play key roles in solving several problems from analysis and geometry, and they also serve as basic models in physics. We will discuss several key properties of non-negative solutions to the elliptic problems, e.g., the sharp conditions governing their existence and their qualitative behavior, while highlighting some of our recent methods for establishing such properties. We shall then motivate the significance of these properties, especially in their connection to some open problems such as the Lane-Emeden and Hardy-Littlewood-Sobolev conjectures.