In recent years the concept and the underpinning technology of the inerter, a mechanical element that resists relative acceleration, attracted significant attention in improving passive dynamic vibration absorbers and energy dissipation devices used for mitigating earthquake and wind induced vibrations to building structures. In the case of dynamic vibration absorbers, inerter devices act as mass amplifiers leading to lightweight and more robust to detuning absorbers. In the case of energy dissipation devices, inerter-like mechanisms act as motion amplifiers leading to more efficient dampers and electromagnetic kinetic energy harvesters. In this seminar, I will first provide an overview of the recently emerged inerter-based vibration absorbers and energy dissipation devices for earthquake and wind engineering applications with an emphasis on the tuned mass damper inerter (TMDI) configurations. I will then discuss, with the aid of illustrative numerical and experimental results, the advantages and open challenges of the TMDI for the seismic protection of fixed based and of base isolated structures, as well as for safeguarding tall building occupant comfort to wind-borne oscillations. Attention will be focused on the salient TMDI features which render meaningful the structural modification and, even, the re-design of buildings for improved performance to dynamic excitations and for structural weight minimization to meet pressing demands for resilient and sustainable structures.