This work investigates the fixed-time distributed coordination control for multiple Euler-Lagrange systems and, in particular, addresses containment control with stationary/dynamic leaders as well as leaderless synchronization control. For the containment control scenario with stationary leaders, the subgraph associated with followers is directed. When dynamic leaders are involved, the information transfer between neighboring followers is bidirectional, for which a novel distributed estimator is developed. For the leaderless synchronization control scenario, the communication network among agents is unidirectional. Three fixed-time distributed control schemes are designed for the aforementioned three cases by applying the fixed-time stability theory. The convergence of the coordination control objectives can be achieved in a fixed time that does not depend on any initial conditions of agents' states, and the settling times are also explicitly derived. Finally, numerical simulations are presented to demonstrate the feasibility of the developed control strategies.