  Ontologies are widely used for representing domain knowledge and meta data,
playing an increasingly important role in Information Systems, the Semantic
Web, Bioinformatics and many other domains. However, logical reasoning that
ontologies can directly support are quite limited in learning, approximation
and prediction. One straightforward solution is to integrate statistical
analysis and machine learning. To this end, automatically learning vector
representation for knowledge of an ontology i.e., ontology embedding has been
widely investigated in recent years. Numerous papers have been published on
ontology embedding, but a lack of systematic reviews hinders researchers from
gaining a comprehensive understanding of this field. To bridge this gap, we
write this survey paper, which first introduces different kinds of semantics of
ontologies, and formally defines ontology embedding from the perspectives of
both mathematics and machine learning, as well as its property of faithfulness.
Based on this, it systematically categorises and analyses a relatively complete
set of over 80 papers, according to the ontologies and semantics that they aim
at, and their technical solutions including geometric modeling, sequence
modeling and graph propagation. This survey also introduces the applications of
ontology embedding in ontology engineering, machine learning augmentation and
life sciences, presents a new library mOWL, and discusses the challenges and
future directions.
