One of the most significant breakthroughs in physics of the last decade has been the discovery that materials with non-trivial topological properties for electronic, electromagnetic, acoustic and mechanical responses can be designed and manufactured at our will through engineered metamaterials (MMs). Here, we review the foundation and the state-of-the-art advances of topological photonics, acoustics and mechanical MMs. We discuss how topological MMs enable nontrivial wave phenomena in physics, engineering, of great interest for a broad range of interdisciplinary science disciplines such as classical and quantum chemistry. We first introduce the foundations of topological materials and the main concepts behind their peculiar features, including the concepts of topological charge and geometric phase. We then discuss the topology of electronic band structures in natural topological materials, like topological insulators and gapless Dirac and Weyl semimetals. Based on these concepts, we review the concept, design and response of topologically nontrivial MMs in photonics and phononics, including topological phases in 2D MMs with and without time-reversal symmetry, Floquet TIs based on spatial and temporal modulation, topological phases in 3D MMs, higher-order topological phases in MMs, non-Hermitian and nonlinear topological MMs and the topological features of scattering anomalies. We also discuss the topological properties emerging in other related contexts, such as the topological aspects of chemical reactions and polaritons. This survey aims at connecting the recent advances in a broad range of scientific areas associated with topological concepts, and highlights opportunities offered by topological MMs for the chemistry community at large.