This work describes the process of creating a corpus of Twitter conversations annotated for the presence of counterspeech in response to toxic speech related to axes of discrimination linked to sexism, racism and homophobia. The main novelty is an annotated dataset comprising relevant tweets in their context of occurrence. The corpus is made up of tweets and responses captured by different profiles replying to discriminatory content or objectionably couched news. An annotation scheme was created to make explicit the knowledge on the dimensions of toxic speech and counterspeech.An analysis of the collected and annotated data and of the IAA that emerged during the annotation process is included. Moreover, we report about preliminary experiments on automatic counterspeech detection, based on supervised automatic learning models trained on the new dataset. The results highlight the fundamental role played by the context in this detection task, confirming our intuitions about the importance to collect tweets in their context of occurrence.