This talk will describe recent well-posedness results for Poisson's equation on R^N with N ≥ 3. Instead of seeking solutions of the equation in standard Sobolev spaces, the problem is posed on spaces where an integrability condition is replaced by a condition of decay at infinity. These spaces will be Banach or Hilbert spaces under natural norms and inner products. Some sharp imbedding theorems will be described that are related to boundedness and intrinsic decay rates at infinity for solutions of Poisson's equation. When N = 3, these provide new estimates for gravitational and electrostatic fields.