Single image super-resolution (SISR) is a notoriously challenging ill-posed problem, which aims to obtain a high- resolution (HR) output from one of its low-resolution (LR) versions. To solve the SISR problem, recently powerful deep learning algorithms have been employed and achieved the state- of-the-art performance. In this survey, we review representative deep learning-based SISR methods, and group them into two categories according to their major contributions to two essential aspects of SISR: the exploration of efficient neural network archi- tectures for SISR, and the development of effective optimization objectives for deep SISR learning. For each category, a baseline is firstly established and several critical limitations of the baseline are summarized. Then representative works on overcoming these limitations are presented based on their original contents as well as our critical understandings and analyses, and relevant comparisons are conducted from a variety of perspectives. Finally we conclude this review with some vital current challenges and future trends in SISR leveraging deep learning algorithms. Deep Learning for Single Image Super-Resolution: A Brief Review