Targeted Sentiment Analysis (TSA) is a central task for generating insights from consumer reviews. Such content is extremely diverse, with sites like Amazon or Yelp containing reviews on products and businesses from many different domains. A real-world TSA system should gracefully handle that diversity. This can be achieved by a multi-domain model - one that is robust to the domain of the analyzed texts, and performs well on various domains. To address this scenario, we present a multi-domain TSA system based on augmenting a given training set with diverse weak labels from assorted domains. These are obtained through self-training on the YELP reviews corpus. Extensive experiments with our approach on three evaluation datasets across different domains demonstrate the effectiveness of our solution. We further analyze how restrictions imposed on the available labeled data affect the performance, and compare the proposed method to the costly alternative of manually gathering diverse TSA labeled data. Our results and analysis show that our approach is a promising step towards a practical domain-robust TSA system.