Non-Abelian-based theories are one of the foundations of modern physics. For example, non-Abelian braiding realized by the dynamic winding ofanyonsis widely considered an important candidate for quantum computers. Operations defined by non-Abelian groups are noncommutative in character, meaning that the outcomes of sequential operations depend on their orders. Non-Abelian processes are mathematically captured by unitary matrices, which can manifest as a Berry-phase matrix that connects holonomic adiabatic evolutions of multiple states in parameter space. Because the Berry phase is pervasive in a wide variety of systems, non-Abelian operations are realizable in classical waves such as sound and light. In this talk, I will present our recent findings in classical-wave non-Abelian operations. I will discuss the realization of non-Abelian braiding in acoustics and photonics. Here, the braiding operations are implemented using coupled waveguide arrays, which are adiabatically modulated to enforce a multi-state Berry-phase matrix that swaps the modal dwell sites. The non-Abelian characteristics are revealed by switching the order of two distinct braiding operations involving at least three modes.