In this manuscript, the topic of multi-corpus Speech Emotion Recognition (SER) is approached from a deep transfer learning perspective. A large corpus of emotional speech data, EmoSet, is assembled from a number of existing SER corpora. In total, EmoSet contains 84181 audio recordings from 26 SER corpora with a total duration of over 65 hours. The corpus is then utilised to create a novel framework for multi-corpus speech emotion recognition, namely EmoNet. A combination of a deep ResNet architecture and residual adapters is transferred from the field of multi-domain visual recognition to multi-corpus SER on EmoSet. Compared against two suitable baselines and more traditional training and transfer settings for the ResNet, the residual adapter approach enables parameter efficient training of a multi-domain SER model on all 26 corpora. A shared model with only $3.5$ times the number of parameters of a model trained on a single database leads to increased performance for 21 of the 26 corpora in EmoSet. Measured by McNemar's test, these improvements are further significant for ten datasets at $p<0.05$ while there are just two corpora that see only significant decreases across the residual adapter transfer experiments. Finally, we make our EmoNet framework publicly available for users and developers at https://github.com/EIHW/EmoNet. EmoNet provides an extensive command line interface which is comprehensively documented and can be used in a variety of multi-corpus transfer learning settings.