Global climate change plays an essential role in our daily life. Mesoscale ocean eddies have a significant impact on global warming, since they affect the ocean dynamics, the energy as well as the mass transports of ocean circulation. From satellite altimetry we can derive high-resolution, global maps containing ocean signals with dominating coherent eddy structures. The aim of this study is the development and evaluation of a deep-learning based approach for the analysis of eddies. In detail, we develop an eddy identification and tracking framework with two different approaches that are mainly based on feature learning with convolutional neural networks. Furthermore, state-of-the-art image processing tools and object tracking methods are used to support the eddy tracking. In contrast to previous methods, our framework is able to learn a representation of the data in which eddies can be detected and tracked in more objective and robust way. We show the detection and tracking results on sea level anomalies (SLA) data from the area of Australia and the East Australia current, and compare our two eddy detection and tracking approaches to identify the most robust and objective method.