Phononic crystals and acoustic metamaterials have found their way into numerous applications ranging from vibration suppression to imaging beyond the diffraction limit. The functional response of these architected materials is typically explained by the band structure which describes the wave dispersion in such media. In recent years, using fundamental concepts from topology – a branch of mathematics, less apparent details in the band structure of phononic crystals and metamaterials have been extracted to explain and explore new acoustics. The first half of the talk will focus on introducing topological acoustics. The means to topological waveguiding and the science behind the phenomena will be discussed. In the second half, the focus will shift towards piezoelectric phononic structures and the tuning of wave dispersion which enables on-demand robust waveguiding through dynamic reconfigurability. We will look at topological electroacoustic lattices which are inspired by the models of 1D and 2D quantum systems but have their own unique features due to the physics of mechanical waves. The use of electrical elements within mechanical systems enables unprecedented dispersion engineering and produces creative platforms to understand novel wave propagation phenomena and develop smart materials with potential applications in fields such as energy harvesting, next-gen communications, soft robotics, and edge computing.