The presence of offensive language on social media platforms and the implications this poses is becoming a major concern in modern society. Given the enormous amount of content created every day, automatic methods are required to detect and deal with this type of content. Until now, most of the research has focused on solving the problem for the English language, while the problem is multilingual. We construct a Danish dataset DKhate containing user-generated comments from various social media platforms, and to our knowledge, the first of its kind, annotated for various types and target of offensive language. We develop four automatic classification systems, each designed to work for both the English and the Danish language. In the detection of offensive language in English, the best performing system achieves a macro averaged F1-score of 0.74, and the best performing system for Danish achieves a macro averaged F1-score of 0.70. In the detection of whether or not an offensive post is targeted, the best performing system for English achieves a macro averaged F1-score of 0.62, while the best performing system for Danish achieves a macro averaged F1-score of 0.73. Finally, in the detection of the target type in a targeted offensive post, the best performing system for English achieves a macro averaged F1-score of 0.56, and the best performing system for Danish achieves a macro averaged F1-score of 0.63. Our work for both the English and the Danish language captures the type and targets of offensive language, and present automatic methods for detecting different kinds of offensive language such as hate speech and cyberbullying.