The recent discovery of topological phases of condensed matter has recently spawned a quest for their classical analogs in other branches of physics. In wave physics, in particular, several proposals have been put forward to obtain artificial periodic materials with a topologically nontrivial band structure, leading to metamaterial analogs of topological insulators. In particular, these materials support backscattering-immune chiral wave transport on their edges, along with a remarkable topological resilience against a large class of defects and disorder. In this presentation, we review our recent theoretical and experimental work on acoustic and electromagnetic topological insulators. We discuss our strategies to break time-reversal symmetry to induce a non-zero Chern invariant in nonreciprocal acoustic metamaterials, and pseudospin engineering methods to induce topologically nontrivial subwavelength acoustic and electromagnetic states without breaking time-reversal symmetry. Our results may lead to a novel class of system that exploit topological protection to guide and manipulate waves in unprecedented ways.