Predicting the performance of engineering systems where large-scale structural failure originates from small-scale phenomena—such as defects, microcracks, or inhomogeneities—remains a grand challenge in computational mechanics. To address this, concurrent multiscale methods are essential for capturing the tightly coupled interaction between small-scale failure mechanisms and large-scale structural dynamics. In this work, we advance the Schwarz alternating method as a flexible, robust, and minimally intrusive framework for concurrent multiscale coupling in both quasistatic and dynamic finite deformation solid mechanics. While the Schwarz method is traditionally viewed as a preconditioner for Krylov subspace methods, we utilize it here as a solution method for coupled partial differential equations. For transient dynamics, we introduce a global time-stepping scheme that allows for the coupling of subdomains with disparate time scales and distinct time integration schemes (e.g., implicit-explicit coupling). Furthermore, we report on current investigations into coupling through a non-overlapping Schwarz formulation. Originally developed for contact mechanics, this approach provides a complementary framework for coupling domains where no overlap exists, contrasting with the classical method's requirement for a geometric overlap region. A significant advantage of the proposed framework is its implementation within massive-parallel codes without requiring invasive modification of the core solvers. We demonstrate the versatility of this approach through a series of numerical examples ranging from fundamental verification tests to industrial-scale applications. We further discuss optimization strategies for the Schwarz method, presenting parameter studies on the trade-offs between overlap volume, mesh refinement, and iteration counts to maximize computational efficiency while maintaining high-fidelity results. Sandia National Laboratories is a multi-mission laboratory managed and operated by National Technology and Engineering Solutions of Sandia, LLC., a wholly owned subsidiary of Honeywell International, Inc., for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-NA0003525.