Recommender systems have shown great potential to help users find interesting and relevant Web service (WS) from within large registers. However, with the proliferation of WSs, recommendation becomes a very difficult task. Social computing seems offering innovative solutions to overcome those shortcomings. Social computing is at the crossroad of computer sciences and social sciences disciplines by looking into ways of improving application design and development using elements that people encounter daily such as social networks, trust, reputation, and recommendation. In this paper, we propose a social trust-aware system for recommending Web services (WSs) based on social qualities of WSs that they exhibit towards peers at run-time, and trustworthiness of the users who provide feedback on their overall experience using WSs. A set of experiments to assess the fairness and accuracy of the proposed system are reported in the paper, showing promising results and demonstrating that our service recommendation method significantly outperforms conventional similarity-based and trust-based service recommendation methods.