Knowledge graphs serve as an essential source for many tasks in data mining, artificial intelligence, and interdisciplinary challenges such as entity disambiguation, fact checking, and computational journalism. However, real-world knowledge graphs are far from perfect, so it remains a crucial task to improve the quality and completeness of these essential tools.In this work, I present a representation learning-based approach to improve the quality of knowledge graphs through a process called Knowledge Graph Completion. This process finds missing connections among entities in the graph. I also propose a relaxed version of the knowledge graph completion task called Open-world Knowledge Graph Completion to complete and extend knowledge graphs with unobserved entities. Extensive evaluations on several data sets with different sizes shows the effectiveness of proposed methods for knowledge graph completion. Further experiments also demonstrate that these models can also solve tasks beyond Knowledge Graph Completion, such as classification, clustering, and recommendation tasks based on the representations learned from the knowledge graphs.