Lightweight face recognition models, as one of the most popular and long-standing topics in the field of computer vision, has achieved vigorous development and has been widely used in many real-world applications due to fewer number of parameters, lower floating-point operations, and smaller model size. However, few surveys reviewed lightweight models and reimplemented these lightweight models by using the same calculating resource and training dataset. In this survey article, we present a comprehensive review about the recent research advances on the end-to-end efficient lightweight face recognition models and reimplement several of the most popular models. To start with, we introduce the overview of face recognition with lightweight models. Then, based on the construction of models, we categorize the lightweight models into: (1) artificially designing lightweight FR models, (2) pruned models to face recognition, (3) efficient automatic neural network architecture design based on neural architecture searching, (4) Knowledge distillation and (5) low-rank decomposition. As an example, we also introduce the SqueezeFaceNet and EfficientFaceNet by pruning SqueezeNet and EfficientNet. Additionally, we reimplement and present a detailed performance comparison of different lightweight models on the nine different test benchmarks. At last, the challenges and future works are provided. There are three main contributions in our survey: firstly, the categorized lightweight models can be conveniently identified so that we can explore new lightweight models for face recognition; secondly, the comprehensive performance comparisons are carried out so that ones can choose models when a state-of-the-art end-to-end face recognition system is deployed on mobile devices; thirdly, the challenges and future trends are stated to inspire our future works.