The study of topology in elastic media has been primarily focused on achieving non-trivial topological states in discrete elastic lattices through active or chiral microscopic interactions. Realization of such topological states in continuous elastic media remains largely unexplored. In this study, a new continuum theory of micropolar gyroelasticity is developed and applied to attain non-trivial topological boundary states in elastic continua. According to the new theory, an elastic continuum is composed of elastically interacting micro-volume elements that can translate and rotate and are connected at their mass centers to gyroscopes, which contribute to the linear and orbital angular momenta but not to the spin angular momentum of the continuum. By applying this micropolar gyroelasticity theory to elastic media with both periodic and finite domains, the emergence of topological boundary states in 2D micropolar gyroelastic continua is demonstrated. Through using the Floquet–Bloch method for periodic domains, the bulk-boundary correspondence is analytically established, and the emergence of non-trivial topological bulk states characterized by Mexican-hat band structures is observed. In addition, by employing an asymptotic analytical model based on the extended Bloch theorem and performing numerical analyses of micropolar gyroelastic continua with finite domains of different geometries, it is shown that the non-trivial Mexican-hat band structure is associated with and provides protection for topological boundary states confined at the boundaries. Finally, the application of the newly developed micropolar gyroelasticity theory to Zinc-blende structured materials (including ZnTe, GaP, InP and ZnS) reveals that the emergence of the topological boundary states in an elastic continuum is not triggered solely by the gyroscopic effect but also depends on the material properties of the micropolar continuum. This study provides new insights into extending notions and methods of topology to analyze elastic continua, paving the way for the practical implementation of topological mechanical systems in various engineering applications. •A new theory of micropolar gyroelasticity is developed to attain non-trivial topological boundary states in continua.•The emergence of topological boundary states in 2D continua is shown for elastic media with periodic and finite domains.•The bulk-boundary correspondence is established, and the emergence of non-trivial topological bulk states is observed.•The non-trivial Mexican-hat band structure provides protection for topological boundary states confined at the boundaries.•The emergence of the topological boundary states depends on both the gyroscopic effect and properties of the continuum.