One key consequence of the information revolution is a significant increase and a contamination of our information supply. The practice of fact checking wont suffice to eliminate the biases in text data we observe, as the degree of factuality alone does not determine whether biases exist in the spectrum of opinions visible to us. To better understand controversial issues, one needs to view them from a diverse yet comprehensive set of perspectives. For example, there are many ways to respond to a claim such as animals should have lawful rights, and these responses form a spectrum of perspectives, each with a stance relative to this claim and, ideally, with evidence supporting it. Inherently, this is a natural language understanding task, and we propose to address it as such. Specifically, we propose the task of substantiated perspective discovery where, given a claim, a system is expected to discover a diverse set of well-corroborated perspectives that take a stance with respect to the claim. Each perspective should be substantiated by evidence paragraphs which summarize pertinent results and facts. We construct PERSPECTRUM, a dataset of claims, perspectives and evidence, making use of online debate websites to create the initial data collection, and augmenting it using search engines in order to expand and diversify our dataset. We use crowd-sourcing to filter out noise and ensure high-quality data. Our dataset contains 1k claims, accompanied with pools of 10k and 8k perspective sentences and evidence paragraphs, respectively. We provide a thorough analysis of the dataset to highlight key underlying language understanding challenges, and show that human baselines across multiple subtasks far outperform ma-chine baselines built upon state-of-the-art NLP techniques. This poses a challenge and opportunity for the NLP community to address.