Fine-grained entity typing (FGET) is an important natural language processing task. It is to assign fine-grained semantic types of a type taxonomy (e.g., Person/artist/actor) to entity mentions. Fine-grained entity semantic types have been successfully applied in many natural language processing (NLP) applications, such as relation extraction, entity linking and question answering. The key challenge for FGET is how to deal with label noises that disperse in the corpora since the corpora are normally automatically annotated. Various type taxonomies, typing methods and representation learning approaches for FGET have been proposed and developed in the past two decades. This paper systematically categorizes and reviews these various typing methods and representation learning approaches to provide a reference for future studies on FGET. We identify the current trends in FGET research: (i) Learning embedded feature representations to address the challenges posed by label noises, tail types and new entities; (ii) Tackling FGET jointly with other entity analysis sub-tasks (e.g., entity linking and coreference resolution) is also a promising direction. We also present a comprehensive review of type taxonomies, resources, applications for FGET and methods for automatically generating FGET training corpora.