Everyday conversation frequently includes challenges to the clarity of the acoustic speech signal, including hearing impairment, background noise, and foreign accents. Although an obvious problem is the increased risk of making word identification errors, extracting meaning from a degraded acoustic signal is also cognitively demanding, which contributes to increased listening effort. The concepts of cognitive demand and listening effort are critical in understanding the challenges listeners face in comprehension, which are not fully predicted by audiometric measures. In this article, the authors review converging behavioral, pupillometric, and neuroimaging evidence that understanding acoustically degraded speech requires additional cognitive support and that this cognitive load can interfere with other operations such as language processing and memory for what has been heard. Behaviorally, acoustic challenge is associated with increased errors in speech understanding, poorer performance on concurrent secondary tasks, more difficulty processing linguistically complex sentences, and reduced memory for verbal material. Measures of pupil dilation support the challenge associated with processing a degraded acoustic signal, indirectly reflecting an increase in neural activity. Finally, functional brain imaging reveals that the neural resources required to understand degraded speech extend beyond traditional perisylvian language networks, most commonly including regions of prefrontal cortex, premotor cortex, and the cingulo-opercular network. Far from being exclusively an auditory problem, acoustic degradation presents listeners with a systems-level challenge that requires the allocation of executive cognitive resources. An important point is that a number of dissociable processes can be engaged to understand degraded speech, including verbal working memory and attention-based performance monitoring. The specific resources required likely differ as a function of the acoustic, linguistic, and cognitive demands of the task, as well as individual differences in listeners' abilities. A greater appreciation of cognitive contributions to processing degraded speech is critical in understanding individual differences in comprehension ability, variability in the efficacy of assistive devices, and guiding rehabilitation approaches to reducing listening effort and facilitating communication.