We present a neural framework for opinion summarization from online product reviews which is knowledge-lean and only requires light supervision (e.g., in the form of product domain labels and user-provided ratings). Our method combines two weakly supervised components to identify salient opinions and form extractive summaries from multiple reviews: an aspect extractor trained under a multi-task objective, and a sentiment predictor based on multiple instance learning. We introduce an opinion summarization dataset that includes a training set of product reviews from six diverse domains and human-annotated development and test sets with gold standard aspect annotations, salience labels, and opinion summaries. Automatic evaluation shows significant improvements over baselines, and a large-scale study indicates that our opinion summaries are preferred by human judges according to multiple criteria.