In this chapter we investigate the elastic response of beams, which provides a means for calculating the lateral load-carrying and deflection characteristics of beams. Beams are among the most widely used structural elements and are often subjected to severe loading conditions which can induce severe vibrations. Two methods (Euler–Bernouli beam theory and Timoshenko beam theory) are addressed as the primary focus of this chapter. However, the discussions begin by developing the equations of motion for a shear beam (one which experiences only shear deformation). Then the equations of motion for simple beam theory, also called Euler–Bernouli beam theory is developed and discussed. The discussions include free vibrations (with the effect of initial conditions considered) and forced vibration. The use of Laplace transformations in solving beam vibration problems is presented, as well as discussions pertaining to the frequency response function. The simple beam theory section concludes by examining the effect of axial loads on the solution. Next, Timoshenko beam theory is considered in which shear deformation and rotary inertia is accounted for. The equations of motion are developed and subsequently, both free and forced vibrations are considered with the effect of shear deformation and rotary inertia for free vibrations are discussed.