Large language models have shown impressive performance on many natural language processing (NLP) tasks in a zero-shot setting. We ask whether these models exhibit commonsense understanding -- a critical component of NLP applications -- by evaluating models against four commonsense benchmarks. We find that the impressive zero-shot performance of large language models is mostly due to existence of dataset bias in our benchmarks. We also show that the zero-shot performance is sensitive to the choice of hyper-parameters and similarity of the benchmark to the pre-training datasets. Moreover, we did not observe substantial improvements when evaluating models in a few-shot setting. Finally, in contrast to previous work, we find that leveraging explicit commonsense knowledge does not yield substantial improvement.