Visual recognition is currently one of the most important and active research areas in computer vision, pattern recognition, and even the general field of artificial intelligence. It has great fundamental importance and strong industrial needs, particularly the modern deep neural networks (DNNs) and some brain-inspired methodologies, have largely boosted the recognition performance on many concrete tasks, with the help of large amounts of training data and new powerful computation resources. Although recognition accuracy is usually the first concern for new progresses, efficiency is actually rather important and sometimes critical for both academic research and industrial applications. Moreover, insightful views on the opportunities and challenges of efficiency are also highly required for the entire community. While general surveys on the efficiency issue have been done from various perspectives, as far as we are aware, scarcely any of them focused on visual recognition systematically, and thus it is unclear which progresses are applicable to it and what else should be concerned. In this survey, we present the review of recent advances with our suggestions on the new possible directions towards improving the efficiency of DNN-related and brain-inspired visual recognition approaches, including efficient network compression and dynamic brain-inspired networks. We investigate not only from the model but also from the data point of view (which is not the case in existing surveys) and focus on four typical data types (images, video, points, and events). This survey attempts to provide a systematic summary via a comprehensive survey that can serve as a valuable reference and inspire both researchers and practitioners working on visual recognition problems.