Face Super-Resolution (FSR) represents a significant branch of image super-resolution, aiming to reconstruct low-resolution face images into high-resolution counterparts. Recently, driven by rapid advancements in deep learning technology, FSR methods using deep learning have achieved notable subjective and objective reconstruction quality, attracting extensive industrial attention. However, detailed classifications of FSR methods remain limited. Therefore, this survey systematically and comprehensively reviews deep learning-based FSR methods. Initially, we introduce the background and technical framework of FSR. Subsequently, we detail the FSR problem definition, alongside commonly used datasets, evaluation metrics, and loss functions. We conduct comprehensive researches in deep learning FSR methods and classify them according to their solution strategies. Within each category, we begin with a general method description, and subsequently introduce representative approaches and discuss their respective pros and cons. Finally, we address current challenges in FSR methods and propose future research directions.