In recent times, verbal aggression and related phenomena of hate speech, abusive language, trolling, etc. have become a major problem over social media. In this paper, I present the results of a large-scale quantitative study of aggression based on a target-based typology in a manually-annotated multilingual dataset of over 20,000 Facebook comments and tweets each written in Hindi, English or code-mixed Hindi-English. Taking insights from this study, I develop 2 different classifiers for detecting aggression in Hindi, English and Hindi-English mixed Facebook and Twitter conversations. The classifiers are developed using an annotatedcorpus of approximately 9,000 Facebook comments and 5,000 tweets. Since a phenomenon like aggression is highly subjective, the study shows a comparatively modest inter-annotator agreement of 0.72 and an overall F1 score of 0.64 for both Facebook and Twitter. Consequently, I also carried out two user studies, where humans were asked to evaluate the annotations by the classifier, to test the actual 'acceptance' of the classifier's judgments. I discuss the results of this user study and give an analysis of the overall performance of the system.