The deep learning revolution is touching all scientific disciplines and corners of our lives as a means of harnessing the power of big data. Marine ecology is no exception. These new methods provide analysis of data from sensors, cameras, and acoustic recorders, even in real time, in ways that are reproducible and rapid. Off-the-shelf algorithms can find, count, and classify species from digital images or video and detect cryptic patterns in noisy data. Using these opportunities requires collaboration across ecological and data science disciplines, which can be challenging to initiate. To facilitate these collaborations and promote the use of deep learning towards ecosystem-based management of the sea, this paper aims to bridge the gap between marine ecologists and computer scientists. We provide insight into popular deep learning approaches for ecological data analysis in plain language, focusing on the techniques of supervised learning with deep neural networks, and illustrate challenges and opportunities through established and emerging applications of deep learning to marine ecology. We use established and future-looking case studies on plankton, fishes, marine mammals, pollution, and nutrient cycling that involve object detection, classification, tracking, and segmentation of visualized data. We conclude with a broad outlook of the field's opportunities and challenges, including potential technological advances and issues with managing complex data sets.