An explicit–explicit staggered time-integration algorithm and an implicit–explicit counterpart are presented for the solution of non-linear transient fluid–structure interaction problems in the Arbitrary Lagrangian–Eulerian (ALE) setting. In the explicit–explicit case where the usually desirable simultaneous updating of the fluid and structural states is both natural and trivial, staggering is shown to improve numerical stability. Using rigorous ALE extensions of the two-stage explicit Runge–Kutta and three-point backward difference methods for the fluid, and in both cases the explicit central difference scheme for the structure, second-order time-accuracy is achieved for the coupled explicit–explicit and implicit–explicit fluid–structure time-integration methods, respectively, via suitable predictors and careful stagings of the computational steps. The robustness of both methods and their proven second-order time-accuracy are verified for sample application problems. Their potential for the solution of highly non-linear fluid–structure interaction problems is demonstrated and validated with the simulation of the dynamic collapse of a cylindrical shell submerged in water. The obtained numerical results demonstrate that, even for fluid–structure applications with strong added mass effects, a carefully designed staggered and subiteration-free time-integrator can achieve numerical stability and robustness with respect to the slenderness of the structure, as long as the fluid is justifiably modeled as a compressible medium. Copyright © 2010 John Wiley & Sons, Ltd.