Knowledge graphs provide structured representations of facts about real-world entities and relations, serving as a vital knowledge source for numerous artificial intelligence applications. This dissertation seeks to extend the scope and provide theoretical guidance for representation learning based query answering on knowledge graphs. The incompleteness of knowledge graphs has recently motivated the use of representation learning models in recent years to generalize from known facts and infer new knowledge for query answering. Despite advances in answering atomic queries by representing deterministic facts within a monolingual knowledge graph, existing models must overcome the following three challenges: (i) they must address the need to incorporate uncertainty information into query answering, which is critical to many knowledge-driven applications; (ii) they must effectively leverage complementary knowledge from knowledge graphs in different languages; (iii) they must be able to embed complex first-order logical queries.In this dissertation, we address the aforementioned challenges and extend the scope of query answering on knowledge graphs through contributions on the following three fronts: (i) To capture fact uncertainty and support reasoning under uncertainty, we propose two knowledge graph embedding models that are capable of encoding uncertain facts in the embedding space. Our proposed models thus learn entity and relation embeddings according to the confidence scores of uncertain facts. We introduce probabilistic soft logic to infer the confidence score to provide extra supervision for training. We also explore using box embeddings to embed uncertain knowledge graphs and imposing relation property constraints to enhance performance on sparse uncertain knowledge graphs. (ii) To effectively combine knowledge graphs in different languages, we introduce an ensemble learning framework that embeds all knowledge graphs in a shared embedding space, where the association of entities is captured based on self-learning. The framework performs ensemble inference to combine prediction results from embeddings of multiple language-specific knowledge graphs, for which multiple ensemble techniques are investigated. (iii) To support answering complex first-order logical queries, we present a query embedding framework based on fuzzy logic that allows us to define logical operators in a principled and learning-free manner, whereby learn- ing is only required for entity and relation embeddings. The proposed model can further benefit when complex logical queries are available for training. As a result of this research we were able to identify some of the desirable properties that embedding models ought to possess and analyze which of the existing models have these properties. Therefore, the results presented in this dissertation advance the state-of-the-art of query answering on knowledge graphs along different axes and provide conceptual guidance for future research in this field.