Peer review is crucial for advancing and improving science through constructive criticism. However, toxic feedback can discourage authors and hinder scientific progress. This work explores an important but underexplored area: detecting toxicity in peer reviews. We first define toxicity in peer reviews across four distinct categories and curate a dataset of peer reviews from the OpenReview platform, annotated by human experts according to these definitions. Leveraging this dataset, we benchmark a variety of models, including a dedicated toxicity detection model, a sentiment analysis model, several open-source large language models (LLMs), and two closed-source LLMs. Our experiments explore the impact of different prompt granularities, from coarse to fine-grained instructions, on model performance. Notably, state-of-the-art LLMs like GPT-4 exhibit low alignment with human judgments under simple prompts but achieve improved alignment with detailed instructions. Moreover, the model's confidence score is a good indicator of better alignment with human judgments. For example, GPT-4 achieves a Cohen's Kappa score of 0.56 with human judgments, which increases to 0.63 when using only predictions with a confidence score higher than 95%. Overall, our dataset and benchmarks underscore the need for continued research to enhance toxicity detection capabilities of LLMs. By addressing this issue, our work aims to contribute to a healthy and responsible environment for constructive academic discourse and scientific collaboration.