Machine learning methods especially deep neural networks have achieved great success but many of them often rely on a number of labeled samples for training. In real-world applications, we often need to address sample shortage due to e.g., dynamic contexts with emerging prediction targets and costly sample annotation. Therefore, low-resource learning, which aims to learn robust prediction models with no enough resources (especially training samples), is now being widely investigated. Among all the low-resource learning studies, many prefer to utilize some auxiliary information in the form of Knowledge Graph (KG), which is becoming more and more popular for knowledge representation, to reduce the reliance on labeled samples. In this survey, we very comprehensively reviewed over 90 papers about KG-aware research for two major low-resource learning settings -zero-shot learning (ZSL) where new classes for prediction have never appeared in training, and few-shot learning (FSL) where new classes for prediction have only a small number of labeled samples that are available. We first introduced the KGs used in ZSL and FSL studies as well as the existing and potential KG construction solutions, and then systematically categorized and summarized KG-aware ZSL and FSL methods, dividing them into different paradigms such as the mapping-based, the data augmentation, the propagation-based and the optimization-based. We next presented different applications, including not only KG augmented prediction tasks in Computer Vision and Natural Language Processing (e.g., image classification, visual question answering, text classification and knowledge extraction), but also tasks for KG curation (e.g., inductive KG completion), and some typical evaluation resources for each task. We eventually discussed some challenges and future directions on aspects such as new learning and reasoning paradigms, and the construction of high quality KGs.