Selective breeding for improved animal resilience is becoming critical to increase the sustainability of production systems. Despite the existence of a genetic component for resilience, breeding for improved resilience has been limited by the absence of a consensus on its definition and quantifying method. In this work, we provide a review of (i) the definition of resilience and related concepts such as robustness, resistance, and tolerance; (ii) possible quantifying methods for resilience; (iii) its genetic background; and (iv) insights about its improvement through selective breeding. We suggest that a resilient animal may be defined as an individual that is able to cope with a perturbation(s) and rapidly bounce back to normal functioning if altered. Furthermore, since challenging conditions lead to trade-offs and, consequently, deviations between basic physiological functions, we suggest using these deviations as indicators for resilience. These resilience indicators may also be used as proxies to study the genetic determinism and background of resilience in livestock species. Finally, we discuss possible strategies to improve resilience and review the implementation of associated genetic markers for resilience indicators in selection schemes.