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Abstract This broad review summarizes recent advances and “hot” research topics in nanophononics and elastic, acoustic, and mechanical metamaterials based on results presented by the authors at the EUROMECH 610 Colloquium held on April 25–27, 2022 in Benicássim, Spain. The key goal of the colloquium was to highlight important developments in these areas, particularly new results that emerged during the last two years. This work thus presents a “snapshot” of the state-of-the-art of different nanophononics- and metamaterial-related topics rather than a historical view on these subjects, in contrast to a conventional review article. The introduction of basic definitions for each topic is followed by an outline of design strategies for the media under consideration, recently developed analysis and implementation techniques, and discussions of current challenges and promising applications. This review, while not comprehensive, will be helpful especially for early-career researchers, among others, as it offers a broad view of the current state-of-the-art and highlights some unique and flourishing research in the mentioned fields, providing insight into multiple exciting research directions. 

Abstract The discovery of topologically nontrivial electronic systems has opened a new age in condensed matter research. From topological insulators to topological superconductors and Weyl semimetals, it is now understood that some of the most remarkable and robust phases in electronic systems (e.g., quantum Hall or anomalous quantum Hall) are the result of topological protection. These powerful ideas have recently begun to be explored also in bosonic systems. Topologically protected acoustic, mechanical, and optical edge states have been demonstrated in a number of systems that recreate the requisite topological conditions. Such states that propagate without backscattering could find important applications in communications and energy technologies. Here, a topologically bound mechanical state, a different class of nonpropagating protected state that cannot be destroyed by local perturbations, is demonstrated. It is in particular a mechanical analogue of the well‐known Majorana bound states (MBSs) of electronic topological superconductor systems. The topological binding is implemented by creating a Kekulé distortion vortex on a 2D mechanical honeycomb superlattice that can be mapped to a magnetic flux vortex in a topological superconductor.