Language models (LMs) trained on large amounts of data have shown impressive performance on many NLP tasks under the zero-shot and few-shot setup. Here we aim to better understand the extent to which such models learn commonsense knowledge — a critical component of many NLP applications. We conduct a systematic and rigorous zero-shot and few-shot commonsense evaluation of large pre-trained LMs, where we: (i) carefully control for the LMs’ ability to exploit potential surface cues and annotation artefacts, and (ii) account for variations in performance that arise from factors that are not related to commonsense knowledge. Our findings highlight the limitations of pre-trained LMs in acquiring commonsense knowledge without task-specific supervision; furthermore, using larger models or few-shot evaluation is insufficient to achieve human-level commonsense performance.