We apply the path-integral method in the coordinate space to the Casimir effect. We consider several examples: the Casimir energy of a dilute dielectric ball with dispersion, the Casimir energy of a polarized particle near a dielectric ball, and the Casimir energy of a polarized particle inside a perfectly conducting wedge-shaped cavity. The renormalization group equation for the Φ⁴ model is obtained in the coordinate space by a new method that emphasizes the relation between the background field method and the Casimir energy.