Single image super-resolution (SISR) is a notoriously challenging ill-posed problem that aims to obtain a high-resolution output from one of its low-resolution versions. Recently, powerful deep learning algorithms have been applied to SISR and have achieved 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 contributions to two essential aspects of SISR: The exploration of efficient neural network architectures for SISR and the development of effective optimization objectives for deep SISR learning. For each category, a baseline is first established, and several critical limitations of the baseline are summarized. Then, representative works on overcoming these limitations are presented based on their original content, as well as our critical exposition and analyses, and relevant comparisons are conducted from a variety of perspectives. Finally, we conclude this review with some current challenges and future trends in SISR that leverage deep learning algorithms.