In this chapter, we will explore an alternative approach to solving boundary value problems in elasticity: energy methods, specifically variational methods. The power of the principles of minimum potential energy and complementary energy is effectively demonstrated through illustrative examples of beam bending and Saint-Venant torsion, showcasing a self-consistent approach to simplifying complex three-dimensional elasticity problems into more manageable two-dimensional or even one-dimensional models. Their direct applications to numerical solutions are illustrated via the Rayleigh-Ritz method.