In this article, the vibration suppression of an Euler-Bernoulli beam system is considered by using the adaptive boundary control technique. The dynamics of the beam are represented by a partial differential equation and four ordinary differential equations involving functions of both space and time. By using Lyapunov synthesis, the robust boundary control with a disturbance observer is first proposed to suppress the vibration and attenuate the effect of the external disturbances. To compensate for both the system parametric uncertainties and the disturbances uncertainties, the adaptive boundary control is developed. With the proposed boundary control, the state of the beam system is proven to be uniformly ultimately bounded and converge to a small neighbourhood of zero by appropriately choosing the design parameters. The effectiveness of the proposed control is successfully verified by numerical simulations.