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Global climate change and its impact on biodiversity levels have made extinction a relevant topic in biological research. Yet, until recently, extinction has received less attention in macroevolutionary studies than speciation; the reason is the difficulty to infer an event that actually eliminates rather than creates new taxa. For example, in biogeography, extinction has often been seen as noise, introducing homoplasy in biogeographic relationships, rather than a pattern-generating process. The molecular revolution and the possibility to integrate time into phylogenetic reconstructions have allowed studying extinction under different perspectives. Here, we review phylogenetic (temporal) and biogeographic (spatial) approaches to the inference of extinction and the challenges this process poses for reconstructing evolutionary history. Specifically, we focus on the problem of discriminating between alternative high extinction scenarios using time trees with only extant taxa, and on the confounding effect introduced by asymmetric spatial extinction - different rates of extinction across areas - in biogeographic inference. Finally, we identify the most promising avenues of research in both fields, which include the integration of additional sources of evidence such as the fossil record or environmental information in birth-death models and biogeographic reconstructions, the development of new models that tie extinction rates to phenotypic or environmental variation, or the implementation within a Bayesian framework of parametric non-stationary biogeographic models. [ABSTRACT FROM AUTHOR]