Knowledge graphs have recently attracted significant attention from both academia and industry. They have been widely applied in many applications such as information retrieval, question answering, and recommendation systems. Most existing knowledge graphs are large-scale, subject to update over time, and highly incomplete. Hence, while crucial, maintaining knowledge graphs to get close to completion is a costly task. The task of automatic reasoning to predict missing facts is usually referred to as knowledge graph completion, which includes two important subtasks link prediction and entity typing. Various embedding-based representation learning models for knowledge graph completion have been proposed in the literature, but they are not explainable. Rules, as explicit knowledge, are more easily understood by humans, providing an interpretable solution to the completion tasks. Moreover, rules can express high-level schema knowledge contained in knowledge graphs. However, it is difficult to craft rules manually to describe the schema of a large knowledge graph, such as Yago and DBpedia. Therefore, automatic rule extraction and its applications in reasoning tasks are useful and important for large knowledge graphs. Various approaches have been proposed to learn rules over knowledge graphs, but most of them can only extract a subclass of first-order rules, where binary predicates are connected to form chain-like rules. In such rules, only binary predicates are involved, but no unary predicates to express type information. Besides, knowledge graphs have rich type information on entities. It is rarely studied how such type information can be utilised in rule learning. Learning more expressive rules with type information can be challenging, particularly when dealing with large-scale KGs. We found that embedding-based rule learning methods have recently been proposed for large knowledge graphs and show promising performance. To address these gaps, we propose TyRuLe to mine typed rules, a new extension of chain-like rules involving explicit types, over large knowledge graphs. TyRuLe adopts a combined approach of a path-based strategy utilising type information to guide rule searching, and an embedding-based strategy to complement the path-based strategy. The learned typed rules have demonstrated their superiority in link prediction. Additionally, rules have been successfully applied in link prediction, resulting in good performance, but few attempts have been made for entity typing. The challenge of rule-based entity typing lies in two aspects. First, entity typing needs rules that derive entity types as unary predictions. Second, entity types are more naturally defined via tree-like rules, rather than just chainlike rules. Hence, both the chain-like rules and typed rules are insufficient for entity typing. This gap motivates us to learn tree-like rules that can be directly used for entity typing. Learning tree-like rules is more challenging than chain-like rules. Chain-like rules are a special form of tree-like rules while extracting the former is already challenging. In this thesis, we propose TreeRuLe to learn treeshaped rules. Three rule selection strategies (including embedding-based measures) are employed for estimating rule quality. TreeRuLe provides an explainable solution for entity typing and also has the potential to explain other classification tasks, such as image classification. Furthermore, the form of tree-shaped rules is close to the schema rules, but there are still some schema rules that cannot be expressed by either typed or tree-shaped rules, such as "men and women are mutually exclusive" and "a single mom is both single and a mother". Such schema rules are more expressive but rarely explored in rule-learning literature for large knowledge graphs. In this thesis, we propose EL-RuLe to learn more expressive rules, EL rules. A new box-based embedding strategy is utilised to score and filter the unreasonable EL rules. EL-RuLe demonstrates promising results for entity typing, outperforming all the rule learners on all the metrics, especially the TreeRuLe. By considering these challenges and proposing novel methods, this thesis makes contributions to rule learning over knowledge graphs and explainable knowledge graph completion.