This paper addresses the adaptive trajectory tracking problem of uncertain Euler-Lagrange systems subject to external disturbances. In comparison with existing literature, our work has its distinctive characteristics. First, a large class of persistent external disturbances with unbounded energy can be accommodated, which can be a combination of finite number of step signals with unknown magnitudes and sinusoidal signals with unknown frequencies, amplitudes and phase angles. Second, asymptotic trajectory tracking can be achieved, in other words, disturbance rejection, instead of disturbance attenuation, can be attained. Third, the proposed control law is sufficiently smooth and its design does not require a prior bound for either external disturbances or unknown system parameters. Last but not least, the obtained result is global in the sense that it does not depend on initial conditions of the system. (C) 2019 Elsevier Ltd. All rights reserved.